What does the ‘-‘ (dash) after variable names do here?

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Variables inside a ${...} are called « Parameter Expansion ».
Search for that term in the online manual, or the actual manual (line 792).
The ${var-} form is similar in form to ${var:-}. The difference is explained just one line before the :- expansion (line 810):

… bash tests for a parameter that is unset or null. Omitting the colon
results in a test only for a parameter that is unset.

Thus, this form is testing only when a variable is unset (and not null), and replaces the whole expansion ${...} for the value after the -, which in this case is null.

Therefore, the ${var-} becomes:

  1. The value of var when var has a value (and not null).
  2. Also the value of var (the colon : is missing!) when var is null:'', thus: also null.
  3. The value after the – (in this case, null '') if var is unset.

All that is just really:

  1. Expand to '' when var is either unset or null.
  2. Expand to the value of the var (when var has a value).

Therefore, the expansion changes nothing about the value of var, nor it’s expansion, just avoids a possible error if the shell has the option nounset set.

This code will stop on both uses of $var:

#!/bin/bash
set -u

unset var

echo "variable $var"
[[ $var ]] && echo "var set"

However this code will run without error:

#!/bin/bash
set -u

unset var
echo "variable ${var-}"
[[ ${var-} ]] && echo "var set"

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