I've read it will solve the transaction volume (and fees) problem and that Bitcoin could be then used for daily transactions.
At the same time, it looks like Lighting network only makes sense if transactions are bidirectionnal. For instance I don't see why it's useful to pay the grocery store or my rent to my landlord, since they won't send me money on their side.
Am I missing something?
Yes, you are.
The lightning network is useful for paying the grocery store or your landlord because you do so on a repeated basis. Instead of making 1 on-chain transaction for every single payment to the grocery store or to your landlord, you can use LN to essentially batch your payments so you are paying them but only need to make 2 on-chain transactions over a course of a year, for example (instead of 12 monthly rent transactions).
Furthermore, your landlord or your grocery store is likely to be a hub with many open payment channels with other people. So if you wanted to pay someone or receive money from someone, instead of making an on-chain transaction to pay that person or making a payment channel with them, you can route a payment through your landlord or your grocery store to that person. For example, say your neighbor wants to pay you some Bitcoin. He can either make an on-chain transaction sending you money, or he can use his currently open LN payment channel with the landlord and route the payment through the landlord. This essentially means that he gives the landlord money over his payment channel with the landlord and then the landlord sends you the same amount through your payment channel with the landlord.
