How does UTF-8 "variable-width encoding" work?

RFC3629 - UTF-8, a transformation format of ISO 10646 is the final authority here and has all the explanations.

In short, several bits in each byte of the UTF-8-encoded 1-to-4-byte sequence representing a single character are used to indicate whether it's a trailing byte, a leading byte, and if so, how many bytes follow. The remaining bits contain the payload.

UTF-8 was another system for storing your string of Unicode code points, those magic U+ numbers, in memory using 8 bit bytes. In UTF-8, every code point from 0-127 is stored in a single byte. Only code points 128 and above are stored using 2, 3, in fact, up to 6 bytes.

Excerpt from The Absolute Minimum Every Software Developer Absolutely, Positively Must Know About Unicode and Character Sets (No Excuses!)

Each byte starts with a few bits that tell you whether it's a single byte code-point, a multi-byte code point, or a continuation of a multi-byte code point. Like this:

0xxx xxxx    A single-byte US-ASCII code (from the first 127 characters)

The multi-byte code-points each start with a few bits that essentially say "hey, you need to also read the next byte (or two, or three) to figure out what I am." They are:

110x xxxx    One more byte follows
1110 xxxx    Two more bytes follow
1111 0xxx    Three more bytes follow

Finally, the bytes that follow those start codes all look like this:

10xx xxxx    A continuation of one of the multi-byte characters

Since you can tell what kind of byte you're looking at from the first few bits, then even if something gets mangled somewhere, you don't lose the whole sequence.