The "gift" (or, bribe), by a bold personification, appears as the powerful "friend at court," who introduces another, and makes him welcome in high places.

A man's gift maketh room for him - It is, and ever has been, a base and degrading practice in Asiatic countries, to bring a gift or present to the great man into whose presence you come. Without this there is no audience, no favor, no justice. This arose from the circumstance that men must not approach the altar of God without an offering.
Potentates, wishing to be considered as petty gods, demanded a similar homage: -
Munera, crede mihi, capiunt hominesquedeosque;
Placatur donis Jupiter ipse suis.
Ovid
"Believe me, gifts prevail much with both gods and men: even Jupiter himself is pleased with his own offerings."

A man's gift (i) maketh room for him, and bringeth him before great men.
(i) Gets him liberty to speak, and the favour of them that are esteemed.

A man's gift maketh room for him,.... Or "enlarges him" (y); brings him out of prison, or, or out of straits and difficulties with which he, has, been pressed; or it makes way for him to a judge, and for a favourable hearing of his cause; or it enlarges his acquaintance, and gains him respect among men;
and bringeth him before great men; it opens a way for him into the presence and company of great men, being a fee to their servants; or with it he procures a place to wait on them. It is not necessary to understand it of a gift by way of bribe; but to introduce a person to another, and render him acceptable, and appease anger; as in the cases of Jacob and Abigail, Genesis 32:20.
(y)

Blessed be the Lord, who makes us welcome to come to his throne, without money and without price. May his gifts make room for him in our souls.

(Compare Proverbs 17:8, Proverbs 17:23). Disapproval of the fact stated is implied.

16 The gift of a man maketh room for him,
And bringeth him before the great.
That מתּן may signify intellectual endowments, Hitzig supposes, but without any proof for such an opinion. Intellectual ability as the means of advancement is otherwise designated, Proverbs 22:29. But Hitzig is right in this, that one mistakes the meaning of the proverb if he interprets מתן in the sense of שׂחד (vid., at Proverbs 17:8): mtn is an indifferent idea, and the proverb means that a man makes free space, a free path for himself, by a gift, i.e., by this, that he shows himself to be agreeable, pleasing where it avails, not niggardly but liberal. As a proverb expresses it:
Mit dem Hut in der Hand
Kommt man durchs ganze Land
[with hat in hand one goes through the whole land], so it is said here that such liberality brings before the great, i.e., not: furnishes with introductions to them; but helps to a place of honour near the great, i.e., those in a lofty position (cf. לפני, Proverbs 22:29; עם, Psalm 113:8). It is an important part of practical wisdom, that by right liberality, i.e., by liberal giving where duty demands it, and prudence commends it, one does not lose but gains, does not descend but rises; it helps a man over the difficulties of limited, narrow circumstances, gains for him affection, and helps him up from step to step. The ā of מתּן is, in a singular way (cf. מתּנה, מתּנת), treated as unchangeable.