ためらい means “hesitation” or “reluctance,” describing the moment when uncertainty makes you stop before taking action.
In the song, the singer says he stopped because of just a little hesitation, then came to understand the pain of love. It is a memorable, emotionally nuanced word that captures the tension between holding back and finally expressing your feelings.
This is a love song from someone who has been watching from the sidelines and has finally decided to speak up. The title line says it all: "花束のかわりにメロディーを" (instead of a bouquet, a melody). The singer has no flowers to hand over, so he offers a song instead, and right after that, "抱きしめるかわりにこの声を" (instead of a hug, this voice). It is a shy person's grand gesture, and I find it really sweet. He admits "いつも遠くから 君を見ていた" (I was always watching you from far away), then asks for one thing: "でも今日は僕を見つめて" (but today, look at me).
The middle gets bolder. "君を愛する為に 僕は生まれてきたよ" (I was born to love you) is a big claim, and later he flips it into "君に愛される為に" (to be loved by you), which is a nice little grammar lesson too: 愛する is to love, 愛される is to be loved. The line learners will hear most is "僕じゃなきゃ駄目で 君じゃなきゃ駄目さ" (it has to be me, and it has to be you). 駄目 means "no good", so literally it is "if it isn't me, it's no good". By the last line the request shrinks to "今夜だけは僕を見つめて" (just tonight, look at me), which somehow hits harder than all the promises before it.