Learn Hardcore Japanese: kenkyuu ha tugi no dankai ni hairimasita.

は marks 研究 as the topic of the sentence, indicating we’re talking about “the research” in general. You could use が (研究が次の段階に入りました) to emphasize that it was the research (and not something else) that entered the next stage, but it makes the sentence feel more like new information or a sudden revelation. With は, you assume the listener already knows you’re talking about the research and you’re just giving an update.

What does the phrase 次の段階に入りました literally mean, and why is there a の between 次 and 段階?

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.”

Why is に used before 入りました instead of another particle like を?

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.

Why is the verb 入りました in the past tense? Isn’t the research still ongoing?

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。

What nuance does 段階 carry compared to English “stage” or other Japanese words like フェーズ?

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.

Could I say 研究は次の段階を迎えました or 研究は次の段階に進みました instead?

Yes, both are possible but with slight nuance:

段階を迎える (“to welcome/meet a stage”) stresses the idea of reaching a milestone.
段階に進む (“to proceed to a stage”) highlights movement or progress.
段階に入る is more neutral, simply stating “entered into.” Choose based on whether you want to focus on reaching (迎える), progressing (進む), or just entering (入る).

Can I drop 次の and just say 研究は段階に入りました?

You need to specify which stage you mean. If you omit 次の, it becomes vague: “entered a stage,” but the reader won’t know which one. Always include 次の (or another modifier) to clarify which stage is being entered.