Playing a zillion games I guess, and doing studies.
How did they train tactics back in the day?

Seriously, in the days up to Morphy and Anderssen great players were thought to have a "chess daemon" (not a personal possession by Satan or his minions, rather an inexplicable talent). So those guys didn't train, they thought and created on the spot new things, and everyone else followed in their wake.

All the stuff people think they are 'doing' by running a computer, these geeks had to do for real, by hand. They'd study games on a wooden chess set and write down variations that occurred to them, trying to see if sacrifices etc were sound. They'd ponder over opening systems with no external aids and databases etc, different indiviuals would come to their own conclusions and select their own personal repetoire.
I Know these things, I'm extremely OLD!
Yeah I was always curious about that, Us, contemporary chess addicts have magazines, books, chess sites, .... What did they have? I don't don't think imagination alone cuts it. I think aliens helped them, from planet chess. " here learn this game, humanoid! It will help us to selfdistruct your planet for us, it's called skewers!"

All the stuff people think they are 'doing' by running a computer, these geeks had to do for real, by hand. They'd study games on a wooden chess set and write down variations that occurred to them, trying to see if sacrifices etc were sound. They'd ponder over opening systems with no external aids and databases etc, different indiviuals would come to their own conclusions and select their own personal repetoire.
Sounds like a great way to become a very strong chess player to me.

Thank you for posting this! I was wondering the same thing. I've always wondered how great players rise. But I think [correct me if I'm wrong] all they did was to play on their chess board. I can't stress how much I like discussing this topic, but love to contribute my opinion for this community!

They took out a chess board and played both sides to review and try out different positions while making copious quantities of notes regarding pros and cons of the positions.
See 14:40 to 15:20 and 18:50 to 19:25 in this video. GM Yasser Seirawan explains how he tried to make 1 position work . . .
There have been books of tactics and endgame studies since the 1700's. Of course they became better and better over time, but ambitious chess players studied chess books 200 years ago. There were just a lot less of them

How did they train tactics back in the day?
By the only method available: Learn and study tactical patterns, known as motifs. This meant memorizing some "constellations of pieces" (Reti) and studying the mechanics and justifications behind them (Lasker, Alekhine). The "justification" was important because unsound tactics opened the doors for the opponent or the post-mortem analysis to show that the player tried "to tell a lie" and was caught, or got away with it but "is a liar nonetheless".
I remember reading a book –as a teen– where the author went into great lengths (some 40 pages) explaining how to identify a correct Greek gift sacrifice and the mechanics to make it work. Using the same methodology, I analyzed a pretty Rook sacrifice from Boleslavsky in a Sicilian game played in the 50s, which 4 years later paid off in the following Caro-Kann.
Nowadays I see amateurs giving advice on how to increase the tactical strength, mostly around solving hundreds of online problems within a few seconds each. So, no serious work on the mechanics, no attention to the justification, no improvement in the precise calculation and hardly any solid memories from the events,
I know there is a lot of emphasis on tactics training in modern chess with all of the puzzle books and chess software such as ct art, etc. but my question is how the hell did guys like anderssen and them train to get their tactical ability to be so sick. ok maybe some were natural if there is such a thing, but i know at LEAST a few had to work on their game... so how did they manage to get so good at combinations. is there another way to train tactical ability other then solvingg puzzles. did they get good just from playing.. anyone have a clue/?