to explain it all mathmatically yet as simply as I can...
Gearbox turns output shaft with front sprocket on it. Each sprocket has a set number of teeth on it, which pulls that number of links passed the sprocket for each turn. Each link can be measured in distance. So if the front sprocket has 17 teeth and each chain link is 10mm, one turn of the front sprocket (17*10) pulls 170mm of chain. If we put a 16 tooth sprocket on the front, one trun will pull 160mm (16*10) of chain past. Also, each link pulled past the front sprocket pulls a link past the rear sprocket. 1 turn at the front (17 teeth) pulls 17 links past the rear sproket, pulling that many teeth with it. As the rear sprocket has more teeth it doesn't manage a full turn. Each turn of the rear sprocket is 1 turn of the rear wheel, and so a distance travelled can be measured. Changing the ratio of sprockets changes the number of times the front turns to turn the rear once, and so changes the distance travelled for each turn of the front sprocket.
My standard setup on the cbr is
17 tooth front sprocket
44 tooth rear sprocket
This gives a ratio of 2.59:1 (44/17) Or it takes 2.59 turns of the front sprocket to get 1 turn of the rear sprocket (and so 1 turn of the rear wheel).
If I put a 16 tooth front sprocket on, the ratio changes to 2.75:1 (44/16) so now the engine has to turn the front sprocket 2.75 time for 1 turn of the rear wheel. Each turn of the front sprocket has moved the bike a shorter distance. The engine has an easier time to move the bike (force = mass * distance) same mass, less distance, therefore less force required (or same force greater accelleration

)
Changing the rear sprocket has a similar effect, but a can be more refined. 1 tooth at the front changes the ratio quite a bit (it's 1 out of 17), where changing at the rear has a lesser effect (1 out of 44).
If I were to change up 1 tooth at the rear it has the same effect of taking one off the front, but to a lesser degree
Standard ratio
17/44 = 2.59:1
down 1 at front
16/44 = 2.75:1
up one at rear
17/45 = 2.65:1
Going the other way can be done, but beware that the engine only has so much power, and it won't always make the bike faster or more fuel economic.
I tried
17/42 = 2.47:1
and the bike was hard to get off the line, wouldn't reach top speed in 6th gear and drunk way too much fuel.