Do I need to do a loan agreement with my brother for tax purposes? Because he will transfer a significant amount to my account, will I need some document to show to IRS as its reason? If yes, then what? Or is it enough if he mentions some things in the transfer memo?
Both of you will need a proper note. It probably shouldn't be a 0% note since your brother will have to pay income taxes as if you paid interest (see the Intuit article on "imputed interest"). Your brother will show the note if asked why he hasn't file form 709 (Gift Tax Return) for this money, and you will show it if asked why you hadn't reported this income. This note should specifically say that the money has to be repaid under specific realistic conditions (otherwise it's not really a loan but rather a gift). You would probably want an attorney or at least a licensed tax adviser (EA/CPA) to advise you here.
Is it legally feasible that my brother directly transfers the money to the LLC for investments in my name?
Yes, but it doesn't matter with regards to how the transaction between your brother and you is structured. To make the tracing easier, I'd suggest to transfer to your personal account and then make a transfer from your personal account into LLC. This will also help avoid complications later if arguments arise with regards to who contributed what.
One benefit of directly transferring the loan proceeds into the LLC is if you want to deduct the interest as "investment interest expense". There are some conditions and limitations, and TCJA affected that deduction as well, but it may still be useful at some point. You can do it with two steps as well, but be careful with tracing. Check with your tax adviser.