Implement boolean function as multiplexer
First let's see how a multiplexor works. In your case you have a 2-1 (or 2 to 1) multiplexor.
So looking at your diagram, the multiplexor has the following:
2 data inputs: 0 and 1 (0 has a 0 connected and 1 has the value of X connected)
1 control input: Y
1 output: X and Y
So when Y is a logic level 0, the value on the input 0 (a logic level 0 in this case) is seen on the output.
And when Y is a logic level 1, the value on the input 1 (a logic level X in this case) is seen on the output.
If you think about it, when Y is a logic level 0, a logic level 0 is always on the output as a logic level 0 is hardwired to this input. So this equates X and Y.
Now when Y is a logic level 1, the output will be whatever logic level X is at.
You should see that whatever logic level Y is, the multiplexor output always equals the boolean value of X and Y.
EDIT
With 2 to 1 multiplexors like you have in your question, the control input (Y in your case) can be two logic levels, a 0 or a 1. What the two inputs labelled 0 and 1 on the multiplexor mean, is that when Y = 0, the output will equal the logic level connected to 0, and when Y = 1, the output will equal the logic level connected to input 1.
Think of a multiplexor working like the switch in the image below:
When Sel = 0, the output will equal the logic level of I0 and when Sel = 1, the output will equal the logic level of I1

Hope this helps!