JavaScript Regular Expressions Basics

First One

• (.) matches and captures a single character to Group 1, so (.)\1 matches two of the same characters, for instance AA.
• In the string, the only match for this pattern is bb.
• By replacing these two characters bb by the Group 1 capture buffer $1, i.e. b, we replace two chars with one, effectively removing oneb`.

Second One

• Again (.) matches and captures a single character, capturing it to Group 1.
• The pattern matches each character in the string in turn.
• The replacement is the Group 1 capture buffer $1, so we replace each character with itself. Therefore the string is unchanged.

Third One

• Here, forgetting the parentheses for a moment, .* matches the whole string: this is the match.
• The quantifier * means that the Group 1 is reset every time a single character is matched (new group numbers are not created, as group numbering is done from left to right).
• For every character that is matched, that character is therefore captured to Group 1—until the next capture resets Group 1.
• The end value of Group 1 is the the last capture, which is the last character c
• We replace the match (i.e., the whole string) with Group 1 (i.e. c), so the replacement string is c.

The details of group numbering are important to grasp, and I highly recommend you read the linked article about "the gory details".

Reference

• Capture Group Numbering & Naming: The Gory Details
• JavaScript Regex Basics
• Backreferences

This is quite simple when broken down:

With "abbc".replace(/(.)\1/g, "$1"), the result is "abc" because:
(.) references one character.
\1 references the first back reference

So what it says is "find 2 times the same letter" and replace it with the reference. So any doubled character would match and be replaced by the reference.