Firstly, you need to draw one big loop through all the given pieces, by connecting their ends together. The connections all have to be unit length. This means that any given end can be connected to at most 7 other ends.
Secondly, the numbers indicate how often the loop winds around that point.
Intuitively, the winding number of a point with respect to a loop is the number of times the path winds around it. Think of it as putting a nail into the paper, and letting the loop fall loose: the number of times it is still hanging around the nail is the winding number. If it falls down completely, the winding number is 0. For mathematicians: the given number is the absolute value of the winding number.
The loop is allowed to cross itself, but it may not have any sharp angles. The puzzle-applet allows only one way to connect two given ends (with distance 1).
Puzzles in this genre
Next to the puzzle you'll see a figure which indicates which connections are allowed from the top-right end of the bottom-left piece. Note that every end can reach at most 7 other ends.
First look at the top ends of the top two pieces. They can only reach each other, as all other ends are too far away. So we can draw that line already. The puzzle applet allows only one way to connect these two ends without sharp angle. The same holds for the bottom ends of the bottom two pieces.
Now look at the "1" to the right of the piece on the right. The only manner this number can come inside the loop is when we connect the right two ends of that piece. The remaining end of the piece in the top right can now only reach one end, and the same holds for the remaining end of the bottom-right piece. This means we can complete the right half of the puzzle.
On the left, the "0" cannot come in the loop, so the leftmost ends of the left piece cannot be connected. The bottom two ends of the left piece also cannot connect, because then the remaining end in the bottom left piece cannot connect to any end. This means that the bottom left end of the left piece connects to the remaining end of the bottom left piece. In the same way, the top right end of the piece on the left connects with the top left piece. Now only two ends remain, which must be connected to each other.
Verify for yourself that all numbers in the solution have the winding number as indicated.