Handicap is designed to get the average of your 'good' rounds. For each round it calculates a 'differential' from the adjusted course rating. So you take your score minus the course rating and then adjust it for the slope by taking the differential and dividing it by the course slope and then multiplying it by 113 (average slope supposedly).
Once you have your differentials you take the best 10 out of your last 20 rounds and average those differentials. You then multiply the result by 95% (don't ask me why) and that gives you the handicap index, which can be used to calculate a course handicap based on the rating/slope of the course you're going to play.
Since it's an average of your best scores, in general you should only beat your handicap about 25% of the time.
The rules of golf have a table that shows how to implement the system if you have fewer than 20 scores. I don't recall what it is right now, but I think you use a higher percentage of your scores early on and then decrease it to 50% over time.
I found a little swing fix this week that has really been paying off. Added a slight pause at the top of my backswing. I went from hitting 6-7 greens a round to hitting 12 on Saturday and 14 on Sunday. Shot my 2 best rounds of the year Saturday and Sunday despite not making anything on the greens. Sunday I shot 78 with 38 putts and no birdies. Mucho room for improvement there.