How does this code move the bird sprite in this sketch?
This part of your code is what changes the sprite's y-value.

y += deltaIde * moveSpeed;


When you click the "jump" button or when the bird falls the y-value is changed to reflect either an upward movement or a downward movement. When your render() method is called the bird will either be drawn higher up on the y-axis (it's flying) or lower on the y-axis (it's falling).

As for the bars, if you watch closely in the video you can see that they aren't actually randomized.

Compound Assignment Operators

+= is the "plus assign" operator. It adds whatever is on the right side to the variable on the left side. Here is a table of other compound assignment operators: https://en.wikipedia.org/wiki/Operators_in_C_and_C%2B%2B#Compound_assignment_operators

int x = 5;
x += 2; // equals 7; equivalent to x = x + 2;
x = 5;
x = x + 2; // also equals 7; equivalent to x += 2;