Speaking on Leicester's official website, Cullen expressed enthusiasm for the team's tactics under Martin.
"I am buzzing to be here and I can't wait to get started," said Cullen.
He noted that the team's strategic direction aligned with his personal strengths.
"When this opportunity came about - and with the size of the club and the project they're getting started with now - I thought it was the perfect fit for me," said Cullen.
The midfielder added that the squad's target approach fits his preferred playing style.
"The way we want to play football suits me really well. Keeping the football, playing in the opposition half, creating chances and scoring lots of goals," said Cullen.
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Following Cullen's departure, Swansea will rely on players such as Leo Walta, Melker Widell, Moussa Yeo, and Elijah Just to compete for positions behind the main centre-forward.
