Application Programme Interfaces or "APIs" are secure connections that allow regulated businesses, like Lumio, to connect to your financial accounts but only with your permission. It is part of the UK Open Banking [link to "What is Open Banking?"] and EU PSD2 initiatives [show "What's the difference between Open Banking and PSD2?" as a linked article]
It's a bit like when you sign in to another app via Facebook or Google, it doesn't change or allow unlimited access to your actual account, and you can stop access at any time from your banking app. You'll get prompted every 3 months to reconnect, so you can never accidentally leave the connection running.