kenkyuu ha tugi no dankai ni hairimasita.

• 次の段階 literally means “the next stage.”
• 次 means “next,” and when you want to turn it into an adjective modifying 段階, you add the genitive particle の. So 次の段階 = “the stage of next.”
• Then に入りました (“entered into”) completes the idea: “entered into the next stage.”

When you use 入る in the sense of “to enter into a state, period, or stage,” the thing you enter takes the particle に. So:
X に 入る = “to enter into X.”
Here, 次の段階 is X, so we say 次の段階に入りました. If you tried を入った, it would be ungrammatical because 入る is intransitive in this usage.

Japanese often uses the simple past to report that something has happened up to now, similar to the English present perfect (has entered). It’s announcing the completion of the action of entering. If you want to emphasize the current resulting state (“is now in the next stage”), you could say:
研究は次の段階に入っている。

• 段階 is a neutral noun meaning “step” or “phase” in a process.
• フェーズ is just the English loan for “phase” and feels more technical or borrowed from business jargon.
• Use 段階 in everyday or academic contexts; フェーズ in very technical documents or when you want that business-like flavor.