What does "for" attribute do in HTML <label> tag?

The <label> tag allows you to click on the label, and it will be treated like clicking on the associated input element. There are two ways to create this association:

One way is to wrap the label element around the input element:

<label>Input here:
    <input type='text' name='theinput' id='theinput'>
</label>

The other way is to use the for attribute, giving it the ID of the associated input:

<label for="theinput">Input here:</label>
<input type='text' name='whatever' id='theinput'>

This is especially useful for use with checkboxes and buttons, since it means you can check the box by clicking on the associated text instead of having to hit the box itself.

Read more about this element in MDN.

The for attribute associates the label with a control element, as defined in the description of label in the HTML 4.01 spec. This implies, among other things, that when the label element receives focus (e.g. by being clicked on), it passes the focus on to its associated control. The association between a label and a control may also be used by speech-based user agents, which may give the user a way to ask what the associated label is, when dealing with a control. (The association may not be as obvious as in visual rendering.)

In the first example in the question (without the for), the use of label markup has no logical or functional implication – it’s useless, unless you do something with it in CSS or JavaScript.

HTML specifications do not make it mandatory to associate labels with controls, but Web Content Accessibility Guidelines (WCAG) 2.0 do. This is described in the technical document H44: Using label elements to associate text labels with form controls, which also explains that the implicit association (by nesting e.g. input inside label) is not as widely supported as the explicit association via for and id attributes,

In a nutshell what it does is refer to the id of the input, that's all:

<label for="the-id-of-the-input">Input here:</label>
<input type="text" name="the-name-of-input" id="the-id-of-the-input">