I don't understand the language spoken in that video, so I can't tell exactly what the speaker means. I think "I'm twenty years old" is perfectly natural. I don't think it's awkward. I think it is slightly more formal than "I'm twenty."

There is nothing awkward at all about saying "I'm xx years old". It's a perfectly normal and natural statement. It's longer, so we tend to use it when we want to add emphasis. For example "Mum, come on! I'm eighteen years old! Surely you can let me stay out late on my birthday!' Here's a tip: You need to take most of these 'Here's how native speakers really speak English' videos with a big pinch of salt. They just want to attract viewers. Another typical myth peddled by these youtubers is "native speakers never say 'I'm fine' ". This is all nonsense.

"I'm 20 years old" is fine and sounds completely natural. I did not listen to the video, but I wanted to answer your question. What reason did the speaker give for this sounding awkward?

Arkward? No. It is slightly more formal and slightly longer, hence a little less common in conversation.