The lyrics of the original German version tell a story: 99 balloons are mistaken for UFOs, causing a military general to send pilots to investigate. Finding nothing but balloons, the pilots put on a large show of firepower. The display of force worries the nations along the borders and the defence ministers on each side encourage conflict to grab power for themselves.
In the end, a cataclysmic war results from the otherwise harmless flight of balloons and causes devastation on all sides without a victor, as indicated in the denouement of the song: "99 Jahre Krieg ließen keinen Platz für Sieger," which means "99 years of war left no room for victors." The anti-war song finishes with the singer walking through the devastated ruins of the world and finding a single balloon. The description of what happens in the final line of the piece is the same in German and English: "'Denk' an dich und lass' ihn fliegen," or "Think of you and let it go."
She does that 'Captain Kirk' rhyme in the English version too though.
The real treat for German listeners is the first verse: ich, mich, dich, and neun-und-neunZIG (zig is pronounced like ich in the main German dialect).
With all of the 'neunundneunzig' (aka 99) repeated throughout the song, the ich/dich/mich/vielleicht rhymes is really a superior start over the English version.
It's a rhyming scheme that cannot be replicated in English at all.
For me, singing the words myself forces me to understand them.
So just sing along. Every word, and understand as much as you can.
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Once you know all the words, then the next step is to learn the grammar and learn how the words work together. If you give it a few months, full understanding will come!