Our Gemara on Amud Beis establishes a rule:

Converts are not accepted in the days of the Messiah.

The Torah tells us (Shemos 18:1):

Yisro, the priest of Midian, Moses’ father-in-law, heard all that God had done for Moses and for Israel, God’s people, how God had brought Israel out from Egypt.

Rashi, quoting the Midrash, explains that Yisro heard something specific that triggered him to convert. He heard about the splitting of the Red Sea and the War with Amalek. 

The simple understanding of this is that Yisro heard of the miraculous victories. The problem with the simple peshat is shall we consider victory over Amalek is such an amazing feat in the league of splitting the Red Sea? The battle itself had ups and downs and the victory was not absolute, as God Himself declares an ongoing war with Amalek, generations into the future. Furthermore, the Midrash Tanchuma (Ki Tetzeh 89) says that the Jews were perceived as militarily unstoppable by other nations until Amalek took them on. The Midrash uses a play on words “karecha ba-derech” which literally means, “came upon you by the way”, but could be read as “cooled you off”. That is, like a scalding hot bath is cooled by the first person who goes into it. For all the above, how then do we understand that Yisro was so impressed with the “victory” over Amalek and chose to convert?

The Bas Ayin on Parashas Yisro explains it based on our Gemara. The reason why we cannot accept converts during messianic times is that the Jews will be too holy and unapproachable. He also explains this on a deeper mystical level. Every convert represents bringing into the fold an aspect of purity that became mixed with the impurity. From a Psychological perspective, one can understand that different nations and cultures can import beneficial character traits that were not indigenous to the Jewish people. Some of the greatest sages were either converts or from families of converts, such as Shmaya and Avtalyon (Sanhedrin 96b), and Onkelos (Megillah 3a).

Now the statement in the Midrash follows a logical order. Yisro was drawn to convert after hearing about the wonders of the Splitting of the Red Sea. However, he was yet unable to approach Judaism until Amalek came along and knocked them down a few pegs. Then, he was able to approach and convert.