焼き付ける literally means “to burn” or “to bake,” but it can also mean to imprint something deeply in your memory, like a permanent image. In the song, 今を鮮明に焼き付けて means “burn this moment vividly into our memories,” giving the fleeting present a powerful, unforgettable feeling.
This is a love song about being broke, and it is weirdly happy about it. It opens with "もう財布の底は 見えてしまったけど" (I can already see the bottom of my wallet), which is a very Japanese way to say the money is gone. The answer to that is the line that keeps coming back: "それさえも 笑い合った それさえも 恋だった" (we even laughed together about that, even that was love). Being skint is not the problem in this relationship. It is part of the fun. With a title like McDonald Romance, you can picture the cheapest possible date and two people who are fine with it.
The singer also knows it will not last. "きっと永遠には続かないの" (surely it won't go on forever) sits right next to "明日には明日の風が吹くでしょ" (tomorrow's wind will blow tomorrow, right?). That second one is a real Japanese idiom, roughly "tomorrow is another day", so do not worry about what has not happened yet. My favourite line is "空っぽの財布に心は踊る" (with an empty wallet, my heart dances), because it is a bit ridiculous and completely sincere. Then comes the grumble everyone has said in a shop: "なぜ欲しいものはこんなにも値がはるのでしょう" (why are the things I want so expensive?). Watch for 値がはる, an everyday phrase for "to be pricey". The fix the song offers is "今を鮮明に焼き付けて" (burn this moment vividly into memory). The literal verb is 焼き付ける, to burn or print an image in, and that is how they plan to keep it.