Fair question. The story comes from later rabbinic commentaries on the Hebrew Scriptures, called "Midrash".
I don't remember any angels singing at the red sea.
The only singing angels I remember from the Bible were singing about the birth of Christ. You probably don't remember that one.
Seriously, is it actually in the Bible, or is this a later made up thing?
Here's the best summary I could find easily on the web. It comes from an April 2003 sermon by Rabbi Melanie Aron. The whole sermon can be found here.
[The story is] found in the midrash and sometimes is printed in Haggadahs [Passover meal prayer books] around the reading of the ten plagues.
Imagine the angels watching anxiously as the Israelites leave Egypt and begin their march towards the desert. The rabbis describe the angels sort of like fans at a ball game, sitting up in the bleachers, watching what's going on, and cheering on their favorites.
Hurry up, the angels urge the Israelites, who are only slowly leaving Egypt.
It's taken some time to get the Israelites moving. Packing up their belongings is a job after all they've been in Egypt for 400 years. There are children to prepare for the trip and old people. The Israelites are unaware of any danger, but the angels can see everything at once, notice Pharaoh regretting his decision to free the people, and calling up his horsemen and chariots to chase after them and recapture them.
Oh no, the angels cry, when they see the Israelites heading off in the direction of the Sea of Reeds, the Pharaoh's chariots will catch up to them from behind. They'll be trapped, the angels moan, they can't move forward into the sea, and behind them is all the might of Egypt.
It's hopeless, they exclaim, there is no way out. The angels join in the cries of the Israelites, who by now have turned around and realize the desperateness of their situation.
Of course you know what happens next. In ancient times, since they didn't have night vision goggles, armies hunkered down in the dark and didn't attack. All night the two groups remain still, the Israelites at the shore of the sea, and the Egyptian army just behind them. Then a Ruach Kaddim, a wind from the east, creates a path through the sea.
Following Nachshon ben Aminadab, the first Israelite courageous enough to step into the sea, the Israelites are able to cross safely, but when the Egyptians follow with their soldiers and heavy chariots, they become stuck in the mud and as the waters come rolling back over them, they drown in the sea.
At that point the angels break out into song, they are so happy, so relieved that the Israelites are finally safe. All that God had done for the Israelites has finally paid off, the Israelites are free at last.
God sees the angel's rejoicing, but God isn't pleased. "My creatures are drowning in the sea", God says, "and you sing songs".
The Midrash tells us that God was not angry with the Israelites for singing and rejoicing at the shores of the sea. The people had just escaped great danger. It was only human that they express their relief and their joy. But the angels were supposed to have a somewhat broader perspective. They should have kept their awareness of the spark of God that is in every person, even the Pharaoh himself.
They should have remembered God's teaching, "it is not the death of the wicked that I seek, but only that he should turn from his evil ways and live."
That story from the ancient Midrash is preserved in our Passover seder rituals even to this day. When we come to the retelling of the ten plagues, we pour some wine out of our cup, or some families take a little bit of wine with their finger at this point. We show God that we understand that our cup of joy cannot be filled to the brim, as long as others, even if they were our enemies, have lost their lives.
So where does that leave us today? From all our various vantage points, we join to pray that the war will end soon. We ask for God's guidance too in the postwar period, understanding the rejoicing of Kurds, Shiites, and others at the fall of their enemies, we pray to find the wisdom to turn enemies into friends, and establish the foundation for a more lasting peace.
For Jews, Midrash is part and parcel of learning Torah.
I leave the question of whether the Hebrew Scriptures, their later commentaries, or the Christian Gospels are "a ... made up thing" for another day.
In all of them, and in the Qu'ran too --- amidst a lot of stuff it's hard to wrap a modern mind around --- are inspired bits of wisdom.
Whether they are G-d-inspired, I am never quite sure.