Learn Hardcore Polish: Znam dobry skrót do pracy, ale zimą wolę główną ulicę. - I know a good shortcut to work, but in winter I prefer the main street.

Here the speaker is familiar with a route, so znam is the natural choice.
You can say Znam ten skrót = I know this shortcut, but Wiem ten skrót is not correct.

Why does it say dobry skrót even though it is the object of the verb?

Because znać takes the accusative, but skrót is a masculine inanimate noun. In the singular, masculine inanimate nouns usually have the same form in nominative and accusative.

So:

nominative: dobry skrót
accusative: dobry skrót

The case is still accusative here, even though the form does not change.

A useful comparison:

Widzę dobry film = same-looking accusative
but
Widzę dobrego lekarza = masculine animate, so the form changes

Does skrót really mean shortcut? I thought it meant abbreviation.