The "d" in dy/dx essentially means "difference". dy/dx is saying "the difference in y per difference in x", so when x changes a certain amount y changes, like how you might express "50 kilometers/hour" to mean how much change in kilometers per every 1 hour.

However, with that km/h example, you don't want to just be taking the average over a whole hour. Within that time period, you might be going faster or slower at different times. In calculus, we're more interested in the change in y at an exact moment. To do this, we essentially separate x (equivalent to time here) into infinitely small periods, until it's so small we can ignore the length of time. This is what the textbook means by "limit".

With the kilometers per hour example, you could split the hour into 2 sections of 30 minutes, maybe in the first you moved 30km (then went slower in the 2nd half of the hour). 30km/0.5h = 60km/h. You keep doing this, splitting up the time into smaller and smaller pieces until it's infinitely small. If you know an equation that describes the distance you've traveled at any particular time, you can find the exact speed at any individual moment using methods based on this idea.

Conceptually, implicit differentiation works because if one side of the equation equals the other side at all values of x, that means that the other side of the equation would have to be changing at the same rate no matter what the value of x is as well. This means that if you can find out the rate of change of the left side of the equation (d(left side)/dx), it will equal the rate of change of the right side of the equation (d(right side)/dx).

When you find both of these rates of change, you'll end up having the rate of change of y with respect to x in the equation for one or both sides (whichever have y in it to begin with), because the amount that the whole side of the equation changes with a change in x is dependent on how much y changes with that change in x.