Why <div class="clear"></div> is used after multiple floating DIVS in a container DIV?

The height of an element is determined by its child elements (unless it is explicitly set). E.g.:

+------A------+
|+-----------+|
||     B     ||
|+-----------+|
|+-----------+|
||     C     ||
|+-----------+|
+-------------+

The height of A is determined by where the lower border of its child C is.

Now, floating elements do not count towards the height of their parents, they're taken out of the regular flow:

+------A------+
+--+---------++
   |    B    |
   +---------+
   +---------+
   |    C    |
   +---------+

The A element is collapsed to a minimal height, because its two children are floated.

Clearing elements are introduced to restore the correct height:

+------A------+
|  +---------+|
|  |    B    ||
|  +---------+|
|  +---------+|
|  |    C    ||
|  +---------+|
|+-----D-----+|
+-------------+

The D element is a zero-height element with the clear attribute set. That causes it to fall below the floated elements (it clears them). That causes A to have a regular child element to calculate its height by. That's the most typical use case at least.

The often better solution to introducing extra HTML elements is to set A to overflow: hidden though. That has the effect of forcing a height calculation, which has the same effect of growing the height to the desired size. This solution may or may not have other side effects though.