The context of the “Hosanna” in the liturgy is Psalm 118:25-26, which was taken up again by the people in Jerusalem when Jesus entered at the week of His death:
Save us, we beseech thee, O LORD!
… O LORD, we beseech thee, give us success!
Blessed be he who enters in the name of the LORD!
… We bless you from the house of the LORD. (Ps 118:25-26)
And the crowds that went before him and that followed him shouted, “Hosanna to the Son of David! Blessed is he who comes in the name of the Lord! Hosanna in the highest!” (Mt 21:9; cf. Mk 11:9; Lk 19:38; Jn 12:13)
The expression “Hosanna in the highest” is peculiar because it’s only partially translated into English; they would have been saying the whole thing in Aramaic or Hebrew (and the Gospel only records what they said in Greek, and the Greek doesn’t translate Hosanna either!). The word “highest” would have been a triple-adjective, “high high high”, just like “holy holy holy” is used for “holiest”, because Hebrew’s comparative and superlative adjectives are merely created by repetition: holy holy = holier, holy holy holy = holiest.
It may mean, “Let there be cries of ‘Hosanna’ even among the angels” – that is, let our own cries for salvation be echoed by the angels around the throne of God.
It may mean, “Let our cries of ‘Hosanna’ reach God in heaven” – that is, let our cries for salvation reach the Most High.
It may mean, “Let salvation come to our holy place” – by means of comparing the earthly Temple to God’s Temple (the highest place).
It may mean, “Let heavenly salvation come”.