19 Grk “so that the whole imperial guard.” The ὥστε (hōste) clause that begins v. 13 indicates two results of the spread of the gospel: Outsiders know why Paul is imprisoned (v. 13) and believers are emboldened by his imprisonment (v. 14).
20 The whole imperial guard (Grk “praetorium”) can refer to the elite troops stationed in Rome or the headquarters of administrators in the provinces (cf. Matt 27:27; Mark 15:16; John 18:28, 33; 19:9; Acts 23:35). In either case a metonymy is involved, with the place (the praetorium) put for those (soldiers or government officials) who were connected with it or stationed in it.
21 Grk “it has become known by the whole imperial guard and all the rest.”
22 Grk “my bonds [are].”

In other versions

so that it has become known throughout the whole imperial guard and to all the rest that my imprisonment is for Christ.

As a result, it has become clear throughout the whole palace guard and to everyone else that I am in chains for Christ.

so that it has become known throughout the whole imperial guard, and to everyone else, that my imprisonment is because I am in Christ.

For everyone here, including the whole palace guard, knows that I am in chains because of Christ.

so that my imprisonment in Christ has become known in the whole praetorium and to all the rest,

So that my bonds in Christ are manifest in all the palace, and in all other places;

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