CGP Grey's Rules for Rulers spells out power structures in authoritarian and democratic countries really well.

If you vote, you are saying "I can support you, or I can support the other guy, but I will support someone" whereas not voting tells politicians you are politically useless, so they won't pay any attention to your needs.

It's a cynical way of looking at it, but if the no. 1 imperative for a politician is reelection, spending time doing things that will get you more votes is better than wasting time pleasing people who probably won't vote anyway.