I found that overall, it takes time and experience. You can't build Rome overnight.
BUT, you can do things to increase your progress. Playing with less tension in your arms, hands, wrists, elbows, shoulders, etc. is fundamental to playing faster.
Also, there are a plethora of exercises you can do. Some people are against them, some people are all for them, but I feel that if someone is at a stage of fundamental development, correct practice and mastery of different exercises are benificial to one's facility. Some say that they hinder musicality, but I highly disagree with that for two reasons: 1) Your repertoire and your exercises are totally different pieces of music. Why would how you play one affect how "musical", "deep", "expressive", etc. you play the other? And 2) You can play anything: Hanon, Czerny, scales, arpeggios, even 5-finger exercises as musically as you want to, and I feel that you should! I don't believe in "purely technical" practice. (Another side-note: I don't even like the general use of the word "technique", but I don't see how much about that I can really do.)
Finally, (in most circumstances) DON'T rush the tempo. I believe that a fast passage should not "feel" very fast, frantic, or rushed. There is a fine line between "urgency" (such as the ending of Chopin's 1st or 2nd Ballades or 3rd movement of Beethoven's "Moonlight" sonata) and "rushing", which sounds more like just trying to play with speed instead of portraying a more overall feeling.
I hope these suggestions are helpful.