Bowen scored the winning goal in the 90th minute against Fiorentina as the Hammers lifted their first major trophy in more than 40 years.
It sparked wild scenes back in London and the next day thousands lined the streets to welcome the West Ham squad home in an open-top bus parade.
Talking about winning the Europa Conference League, Bowen said on Rio Ferdinand’s VIBE with FIVE: "It's a bit of a 'pinch me' moment because you don't realise how special something is until you win the trophy.
"And then you come home to Prague to see that parade. You can't put it into words... all the people lining up on top of house, on top of traffic lights to get two minutes with us."
He added: "I didn't realise how big the club was! There are fans everywhere. Where I live everyone is a West Ham fan, no Tottenham fans - or maybe they don't want to tell me they are a Tottenham fan!
"West Ham fans everywhere. I didn't realise how big the club was when I first came, but now I am starting to become one of them, I'm starting to get a feel for it and go 'right, this is a massive club'."
Bowen scored six goals and registered five assists in 38 Premier League appearances last season. With just six games under his belt this term, the forward already has four goals and one assist.
He has set a target of "double figures goals and assists" for himself this season as he looks to emulate the likes of Mohammed Salah and Bukayo Saka.
"I've got to be the one who creates and scores goals, and that's what's expected of me. Last season was not my best season, which people may forget because of the [Europa Conference League-winning] goal, but in terms of my standard I was not at a level I expected of myself.
"Confidence, as a forward player, everything you touch goes in. When those things are going for you, they are going for you so confidence is massive. I want to be the one who steps up and decides games.
"I'm always looking double figures goals and assists, 20-30 goal involvements a season. You look at Salah, Saka and [Riyad] Mahrez, their numbers are the benchmark for me."