Well it depends if they genuinely think they can still win, either because they think they are marginally better or just because they think there is still a chance to outplay you from a dead even position. Also there can be emotions at play - if you have been better for a long time but lost the advantage it can be hard to accept that you have thrown it away.
Saying no to a draw
Agree with Patzer. I am usually so far in front on time (10 min games) that I rarely accept. I have if say we just started and opponent has to go.


Asking for a draw when you know it's over is usually a coward's way out.
It happens all the time in Bullet that people will ask for a draw when they know the position is lost.

Asking for a draw when you know it's over is usually a coward's way out.
It happens all the time in Bullet that people will ask for a draw when they know the position is lost.
I have seen people use this, especially in Bullet, as a tactic to distract and cover up their opponent's time clock after every move

Maybe this is a stupid question but I just lost a game in witch me and my opponent both only had a king and a rook. He would not accept draw and I ran out of time (I had 35 sec left and he had 2 min) Is there no way I can draw in this situation? I tried repeating moves but it did nothing (since he played differently...

I tend to ask for draw if the game is boring. It does annoy me that most players just want to win on time or boring their opponents.

Whilst draws are very very common at the top level it is rarely the expected result at the level > 95% of people on this site play at. Someone or other usually plays an ending badly and loses, unless it is just a trivial draw. I sometimes offer a draw if I find the position boring. Usually I lose soon after they decline cos, quite frankly, I cease to care and don't want to waste any more of my time.
I have very very rarely been offered a draw in a Blitz game though. I've also had players who decline my draw offer and then they lose on time even though they have 3 minutes left and I have < 1 when I offer the draw. If they want to try to flag me its their choice, but they better make sure they make good enough moves fast enough!
I was thinking that people rarely consent to a draw when they are asked, It's like they need those points so desperately. It seems that chess players are kind of angry a lot of the times they play. After a decent game, both sides tried their best, why not say a draw is fine and go for a draw, if asked? I mean it's a game after all, not anything more. Both sides end up about where they started.