You don't have do anything. That transaction will confirm in due course of time. You don't have to keep electrum or even your PC running.

On average it takes 10 minutes for the first confirmation. More confirmations happen all the time as the blockchain gets extended.

Note that you bought 0.001 bitcoins. The unit of account displayed there is mbtc which is 1/1000th of a bitcoin. If you want to change it to btc you can do so via tools > preferences > appearance tab > base unit option.