As it states, "odds" is the ratio that compares the number of favorable outcomes to the number of unfavorable outcomes. Or:

Odds = Number of favorable outcomes

Number of unfavorable outcomes

As opposed to probability:

Probability = Number of favorable outcomes

Number of total outcomes

Given that, many times, especially here, the number of total outcomes = number of favorable outcomes + number of unfavorable outcomes, then the number of unfavorable outcomes = the total number of outcomes - the number of favorable outcomes. So, that would give us:

Odds = Number of favorable outcomes

Total number of outcomes - Number of favorable outcomes

For this problem, on a 6 sided die, there is only one "6". So, there is only one favorable outcome, the single "6" out of 6 total outcomes. So, plugging that into the equation:

Odds = 1/(6-1) = 1/5

Your textbook tells you that the odds are:

number of favorable outcomes / number of unfavorable outcomes.

To get a 6 in a six-sided die, the number of favorable outcomes is 1.

The total number of outcomes is 6. So the number unfavorable outcomes is 5.

The odds are then 1/5

There is only 1 favorable outcome which is 6 and 5 unfavorable outcomes. This means that the odds of a 6 is 1/5