Frank Lampard was speaking after his side's 4-0 defeat by Minnesota United on their pre-season tour in the United States.
Everton secured Premier League safety last season with a win against Crystal Palace in their penultimate game.
"The players have to understand that we were in a relegation battle for a big portion of last season," said Frank Lampard to BBC.
"We had an amazing night [against Palace] and an amazing run which was great historically for the club but, as soon as that finished, I put it to bed very quickly so the players better put it to bed as well.
"We were in that fight for a reason and, if we don't want to be in that situation again, they have to be better, I have to be better."
Everton's defeat by Major League Soccer side Minnesota United followed a 2-0 loss to Arsenal in the first game of their US tour.
They play at Blackpool and host Ukrainian side Dynamo Kyiv in further pre-season games before starting their league campaign at home to Chelsea on 6 August.
Lots to think about, lots of work to do," said Lampard. "What matters is where we are at in two and a half weeks.
"If there is to be good to come out of this, the players and the dressing room have to understand that they have just put in a very poor performance individually and collectively and been well beaten."
Everton have sold winger Richarlison this summer and signed defender James Tarkowski on a free transfer after his contract with Burnley expired.
Frank Lampard says "there is a void in forward areas", while "through the team, there are things that I want to strengthen".
He added: "The lads have trained well out here. They have trained hard as they should do but that performance should give them a real something to think about on the plane home because it has certainly given me something to think about."