The Warriors have home advantage for Saturday's quarter-final against Toulon at Scotstoun, where they are unbeaten this season. The French side are three-time winners of the competition but currently sit 11th in the Top 14.
Victory would secure a semi-final against either Leinster or Sale Sharks at Murrayfield on the first weekend of May.
Head coach Smith highlighted the importance of maintaining composure while also recognising the role of the home support.
"I think we're already emotionally charged for this one," said Smith in The Scotsman. "I think the crowd is going to play a massive role to remind us the whole time we're on the pitch that there's a big purpose in what's happening in the game."
While Glasgow Warriors operate in a city dominated in sporting terms by football, the current landscape offers an opportunity for the club to gain greater prominence.
"Part of the motivation is to represent Glasgow Warriors, the city of Glasgow, the Warrior Nation as we call them," added Smith.
Smith also pointed to the lessons learned from last season's quarter-final defeat away to Leinster and the value of playing at home.
"The fact that we're at home and we believed in this from the first day and now that it's happened. Last year we learned a hard lesson in the quarter-final away from home [losing heavily to Leinster in Dublin] and therefore we know how much it means to have home advantage. And hopefully we can get the benefit of that with the support for the players. That's the motivation."