In order to understand why crypto wallets are needed, we need to look at the traditional approach of moving money between two people using "bank accounts"

To keep track of each client's money, a bank creates a "ledger" that keeps track of balances based on "account numbers." This ledger is kept by an Administrator. It is the Administrator who assigns account numbers to clients, and clients make payments by sending a request to the Administrator. 

In order for the Administrator to fulfil this request,  there needs tobe some sorts of authentication process where the client proves that he/she is the actual account owner. So the bank most likely creates a username and password combination that's tied to each account. 

Relying on an administrator creates the problem of Centralization. You're not in control of your account,  the Administrator is. This is the main problem with the decentralized digital currencies.

The Administrator can deny access to your account, can block or censor transactions,  or even not allow you to open an account at all.

For decentralized currencies,  any one can open an "account" and be able to transact with no one's permission. Here, the Administrator is eliminated.

But, is there a way in which you can create an account without an administrator? Yes! In Crypto,  instead of account numbers,  we use "addresses"

How can one get an address?

All you need is a wallet. With the walle software in your phone, or PC, you can create as many addresses as possible without asking an administrator for an account number.  This also takes few seconds and very time consuming. 

Crypto wallets give regular people the ability to conduct "permissionless transactions" without having a third party involved and without the need to register. 

Wallets also make crypto more accessible to more people than banks.As long as you have a stable Internet,  you can download a crypto wallet, install it, create a crypto addresse and you'll be good to go.