LIFT ME UP= To lift is to elevate, to push or take upwards. In a figurative sense, if I'm down or feeling down, I'm depressed, so if you lift me up, you cheer me up, you save me from depression and I feel good and strong again.

MY HEART IS HEAVY= I feel really sad; big problems are troubling me.

THE HURT= The suffering. A deep hurt is a big suffering.

WHEN I FEEL LIKE= When I want to...
This is followed by –ING because after prepositions and conjunctions we use the verb in the –ING form.

GIVE UP= Surrender, stop trying.

WHEN I HIT THE GROUND= If you fall down (because you want to or because you tripped), the moment your body gets in contact with the ground we say that you hit the ground. In this song, the idea of falling down is metaphorical, it means that you feel really depressed and weak, so when you hit the ground is the moment when you feel all those problems are just too much for you. At that point, some people fall into a deep depression or do something desperate, like going suicidal.

WEAK= The opposite of "strong".

WRAP= /ræp/ If you wrap something, you cover it everywhere with something else (for example you wrap a present with colourful paper). If your arms wrap around me, you embrace me, you hold me tight.

CATCHES ME= If you catch something, you take it while it is moving, you trap it. If I am falling and you catch me, you take me and stop me from falling further down. So when he says that God's love catches him, he must be moving somehow. The context makes it clear: he's moving downwards because he's falling, so what he means is that God catches him when he falls, so he won't break to pieces.

YOUR HEART= Your love.

CARRIES ME= To carry is to take, to transport. If your love carries me, your love gives me the strength I need to continue with my life with all its good and bad things.

I'M LETTING GO= If you let go, you abandon yourself to something more powerful than you, you stop worrying and trust that God will take care of you, you relax. Literally, to let go means to stop holding on to something.

LET YOU DOWN= If you let somebody down you disappoint them, you fail to meet their expectations or you betray them.

SURROUNDED= If you are surrounded, your enemies, problems or whatever are all around you, everywhere.

CRYING OUT= To scream or shout aloud, esp in pain, terror, etc., or simply shouting a cry (aaaaaaah!!)

THE DAWN IS BREAKING=  The dawn is starting. The dawn /dɔ:n/  is the moment when the night is over and the light of the day begins to appear on the horizon.  When the dawn breaks, the daylight is beginning. Also, poetically (as here) it means that a new phase in your life is beginning, full of hope and expectations, and the past is left behind.

OVERTAKEN= You feel overtaken when something that you didn't expected suddenly happens, so you're shocked.