Grudem makes the point in his commentary that this is a verb in the present tense, and that it signals a strong desire. It is interesting that this comment by Peter follows on from his description of the Christ-centred ministry of the prophets, focussed in the Messiah's sufferings and glory.
What do they long to look into?
We can say it is this, they long to look into the person of Jesus and his great work of saving his people.
Of how he as the God-man suffered for his people.
Of how he saved them at the cross by shedding his blood.
Of how he calls them to himself by the sound of the gospel and the work of his Spirit.
Of how he gathers them, guards them, sustains them in faith, makes them more pure and holy like himself, and keeps them believing.
Of how he comforts them in their sorrows, cleanses them when they fall into sin and restores them when they return to him.
Of how he doesn't abandon them in death but has given them a living hope through the resurrection.
Of how he perfects their souls at death and will raise and transform their bodies on the last day.
And of how he will welcome them into his everlasting kingdom to endlessly enjoy God forever.
We can also say this, they long to look into something that wasn't for them.
The holy angels have no sins to be forgiven, they need no Saviour. It is a solemn thought that God did not send a Saviour for the angels who disobeyed and fell. Peter says in his second letter that they await judgement (2 Peter 2:4). The Son of God, says the writer to the Hebrews, does not help angels, but us (Hebrews 2:14-16).
If they long to look into these things, how much more should we for whom the Son of God took on flesh and blood, and for whom he suffered in agony and rose in glory?
Our thoughts must be filled with the glorious person and work of of our Lord Jesus Christ.