Why is it impossible to know precisely the velocity and position of an electron at the same time?

It has to do with an electron having properties of both a particle and and wave.We may be able to find one, but in the process, the other will be changed.

Hope this helps :)

This phenomenon is called the Uncertainty Principle, and there is a fancy equation for it (that doesn't really help you understand why).

I'll try to paint a picture.
When a baseball gets thrown, your eyes "see it" because visibile light (350-750nm...low energy) light hits it and then bounces off it and hits your eyes. Light is fast, so the delay is nothing. You see it moving across the room, and your brain has the ability to predict where it will be at a given moment. This is kind of the basis for how we play catch with a baseball....none of us watch the ball hit the mitt...we just watch it for a moment and we already have its trajectory and destination basically figured out - put the mitt out and catch it.

A baseball is not an electron - it is much smaller. And since it is so small, you can't "see" it with visible light. You can only "see" it with gnarly light, like gamma rays. Gamma rays are really high energy (short wavelength - that is how the "see" an electron). The gamma ray bounces off the electron and then can be deflected toward an instrument. You've just figured out the position of your electron. But here is the problem:

Lets pretend that electron is still the baseball. BUT, the light you use to see it with is a basketball thrown at it. So - here is how you see - you throw a basketball at something, the basketball hits it and bounces to you and hits you in the head...and you say "oh, there is the baseball" (just like the light in the above example).

Well, when the baseball that is flying toward you gets hit with a basketball, the baseball DOES NOT continue on the path it was on, but rather goes flying off in some unknown direction (since the basketballs imparts a lot of energy into the baseballs and changes its momentum). So, you know the position of the baseball, but you have no idea about what you just did to its momentum.

Since you can only "see" electrons with super high energy light, the "seeing" process actually changes the momentum of the electron and sends it flying. So you can't know both position and momentum with accuracy.