The main goal for black is to prevent the Philidor position when the white king is in front of the black pawn which is a well known draw. As an 1100 rapid you most likely don't know what it is and why it's a draw thus I'd offer you to learn it. Understanding that white want to play Kd1->Ke1 if we don't stop, we should do something about it. Now let's have a look at our options.
1. If Ke3 when white has Ra3+ and we have to go back or allow a perpetual checks from the side.
2. If pawn to e3 then Kd1 and we can't stop Ke1 with a draw.
3. If Rh1+ then white king is getting to e1 via d2.
4. If nothing move the Kd1-->Ke1.
But we have Kf3 which prevents white's blockade.
There are various rules of thumb in play. The first is "King Safety". Ke3 is the safest spot on the board for the Black King.
After this, White loses if White loses the Rook, while Black can still win if Black loses the Pawn, so long as White immediately loses the Rook. So Ke3 looks like the start to a small dance. Rd2 also helps to keep White the Rook off that file and it is protected by the King. Stay near the Pawn and remember the Check ability of the Pawn.
That's as much as I can see. By the way, you shouldn't be checking engines during a game. In a Daily correspondence game we're allowed to check for records of previous games. I do that all the time, but engine use can get your account closed.
Even actively coaching like this, mid-game, in a game for points is frowned upon. Coaching is allowed but we should be using the classroom, not the live board, mid-game.
@Whisper, this was a 15|10 game played on my alternate account (to play rapid games instead of classic games in the rapid range). I won the game because the opponent blundered, it would have been a draw otherwise.
Am I not allowed to check the engine for any other game while I have ongoing daily games?
@50kg, thank you, I will read your comment quite some times and review the Philidor position. This is one of those thing whose existence I know, but that I never investigated much. I will probably try to figure out everything slowly with your explanation and some more diving into the Philidor position. If I grasp something but not everything I will come with further questions. Thanks for the help!
Summing up again with my own words the Philidor position...
Philidor position is useful to draw in rook endgames when the opponent is up a pawn.
I have started checking this video:
There is a great chess.com lesson on the Philidor position. You'll even be given 5 exercises to reenforce your understanding.
As @RobRam says it clearly here
"cut the king off and wait for them to push the pawn".
And that is what I have to avoid at all costs as black. It means then... The king must not move to e1, the rook must not have easy life if they go to the third rank.
So, when White reaches the Philidor position it is relatively easy to bring home the draw, it is not a supersharp game. What I feel sharp is preventing it!
And now, in another video, there is the trick... White wants to reach the Lucena position! Not only I have to understand the Philidor position, but the Lucena position as well. It looks like with this question I opened a big can of worms!So, linking also a short video again from ChessDiagnostic about the Lucena position. In truth, here I am jotting down my way to study and understand this endgame. This video is not helping me much understanding and memorizing the Lucena position, to be honest. I will need another video probably.
It sounds like this is enough for a comment. I will go on in further ones. Sorry for the monologue, if it is disturbing anybody please let me know... and if you have some stories about what made you understand such endgames, please feel free to share! 
There is a great chess.com lesson on the Philidor position. You'll even be given 5 exercises to reenforce your understanding.
The problem here is that I have to prevent the Philidor position from happening. I will check the class anyway.
I wonder if this time would be better invested playing than studying videos.
https://www.chess.com/lessons/endgame-patterns
I did the distant opposition and the triangulation, did not go through the Lucena and the Philidor position since I even did not get well the triangulation to be honest. I usually leave courses when there is a concept that is too hard, thinking that the next one will be even harder.
I am really wondering what makes things click.
To put this in context of the ChessDojo training program, the basics of the Philidor & Lucena positions are introduced in the 1400-1500 cohort's endgame training, not earlier. You (and I) are many cohorts below that range, so maybe we are better off grasping simpler endgame concepts like rook vs lone pawn (1100-1200 cohort) or opposite color bishops (1200-1300 cohort).
There are various rules of thumb in play. The first is "King Safety". Ke3 is the safest spot on the board for the Black King.
Bad advice. King safety does not apply in the endgame. In the endgame activate the king, don't hide.
As it was mentioned, only 1 move wins. It's not Ke3.
OK, by knowing a bit about the Lucena position (what was said in the chess.com lesson) and the Philidor position, I was able to win the game playing against the engine a couple of times, but I still do mistakes there. If I got it right:
Kf3 kills two birds with one stone. It prevents the White king to go to Ke1, and the White rook cannot give an effective check with Ra3 since the pawn can defend the king.
The goal for Black is themselves to have the king in the promotion square e1.
@LutjeK, right, these are difficult concepts. Maybe I will not be able to put them in practice in a game, as I will probably not be able to see a Legal's mate since I do not like to play gambits and traps and do not know well how to refute them, but at least recognising them when I see a position and its solution could help.
I will have to check the rook VS lone pawn.
I never subscribed to chessdojo, only did some kind of trial with limited material and no chance to see what is in other cohorts.
Lately I am in general focusing on endgames, since I do many mistakes there and I do not play enough of them since I play slow and I have no time to think when I reach the endgame, I am in deep time trouble at that point.
@amchess16, funny enough, Philidor and Lucena position classes are a breeze compared to the triangulation concept.
To be honest, classes here miss some content for low rated players. The most of them teach very complicated concepts, or start with some beginner things but soon go up to explain super hard things. For example, there are no drills for some basic checkmates like K+Q VS K+B or K+Q VS K+N that are not so obvious to win.
@magipi, I am so bad at activating my king in the endgame. I am too much of a shy player, while in chess you have to attack whenever you can!
Even if at times you have to be patient and wait for the opponent to do a mistake...
@JTZ01, this is why a bit of endgame tactics can help so much... the same positions can happen in the endgame over and over! again!
And such positions give us the chance to apply some concepts with real games!
I must thank all people around here that helped me understand what happened in the position. I will post again when I do not understand things!
It's definitely not an only move. There are many ideas here. Ideally you would trade off the rooks and land in a known winning position/structure that you have memorized. But white will probably decline at every chance.
This is my go to structure. If you get to this position it doesn't matter whose turn it is to move. Black is winning because white can not stop white from promoting.
I meant that you win only with Kf3, @RedMountainFox. Then there are quite some ways to win, but they mostly depend on what White does in the next move. The given position does not end up with the K+P VS K structure you give... at least I never got it while trying to solve it while using concepts from the Lucena position.
By the way, the example you gave is super useful. I am probably not that good at making a position converge where I want to.
