In the Family Midrash Says it is written that Shimshon (Samson) asked his parents to convert the Philistine woman so he can marry her. How can we explain this when in our days we don't convert anyone for marriage?
It is difficult to discuss this (particularly online) with the proper reverence and understanding of Shimshon. This question requires a lot more discussion and explanation than can be given here. However, for a brief and superficial answer,it is not so simple that had he asked Beis Din they would have allowed him to do it.
The verse subsequent to the verse of Shimshon asking his parents to marry a Philistine woman says (Shoftim 14:4):
וְאָבִ֨יו וְאִמּ֜וֹ לֹ֣א יָדְע֗וּ כִּ֤י מֵֽיְהֹוָה֙ הִ֔יא כִּֽי־תֹאֲנָ֥ה הֽוּא־מְבַקֵּ֖שׁ מִפְּלִשְׁתִּ֑ים וּבָעֵ֣ת הַהִ֔יא פְּלִשְׁתִּ֖ים מֹשְׁלִ֥ים בְּיִשְׂרָאֵֽל
His father and mother did not know that it was from the Hashem, that he sought a pretense against the Philistines; now at that time the Philistines were ruling over Israel.
There are various explanations in the Rishonim of what "it was from the Hashem" means but all agree that he converted and married them as an effort to save Klal Yisroel from their Philistine oppressors. As is the simple meaning of the posuk.
Even so it was not a valid conversion, or, presumably marriage.
The Rambam in Issurei Biah 13:14 says:
אַל יַעֲלֶה עַל דַּעְתְּךָ שֶׁשִּׁמְשׁוֹן הַמּוֹשִׁיעַ אֶת יִשְׂרָאֵל אוֹ שְׁלֹמֹה מֶלֶךְ יִשְׂרָאֵל שֶׁנִּקְרָא יְדִיד ה' נָשְׂאוּ נָשִׁים נָכְרִיּוֹת בְּגֵיוּתָן. אֶלָּא סוֹד הַדָּבָר כָּךְ הוּא. שֶׁהַמִּצְוָה הַנְּכוֹנָה כְּשֶׁיָּבֹא הַגֵּר אוֹ הַגִּיֹּרֶת לְהִתְגַּיֵּר בּוֹדְקִין אַחֲרָיו שֶׁמָּא בִּגְלַל מָמוֹן שֶׁיִּטּל אוֹ בִּשְׁבִיל שְׂרָרָה שֶׁיִּזְכֶּה לָהּ אוֹ מִפְּנֵי הַפַּחַד בָּא לְהִכָּנֵס לַדָּת. וְאִם אִישׁ הוּא בּוֹדְקִין אַחֲרָיו שֶׁמָּא עֵינָיו נָתַן בְּאִשָּׁה יְהוּדִית. וְאִם אִשָּׁה הִיא בּוֹדְקִין שֶׁמָּא עֵינֶיהָ נָתְנָה בְּבָחוּר מִבַּחוּרֵי יִשְׂרָאֵל.
One should not think that Samson who saved the Jewish people, and Solomon King of Israel, who is called "the friend of God,"married gentile woman who did not convert. Instead, the matter can be explained as follows: The proper way of performing the mitzvah is when a male or a female prospective convert comes, we inspect his motives for conversion. Perhaps he is coming for the sake of financial gain, in order to receive a position of authority, or he desires to enter our faith because of fear. For a man, we check whether he focused his attention on a Jewish woman. For a woman, we check whether she focused her attention on a Jewish youth etc.
Even so the Rambam continues (Ibid 15)
וְכֵן שִׁמְשׁוֹן גִּיֵּר וְנָשָׂא. וְהַדָּבָר יָדוּעַ שֶׁלֹּא חָזְרוּ אֵלּוּ אֶלָּא בִּשְׁבִיל דָּבָר וְלֹא עַל פִּי בֵּית דִּין גִּיְּרוּם חֲשָׁבָן הַכָּתוּב כְּאִלּוּ הֵן עַכּוּ''ם וּבְאִסּוּרָן עוֹמְדִין. וְעוֹד שֶׁהוֹכִיחַ סוֹפָן עַל תְּחִלָּתָן שֶׁהֵן עוֹבְדוֹת כּוֹכָבִים וּמַזָּלוֹת שֶׁלָּהֶן
Samson converted [women] and married them. It is well known that they converted only because of an ulterior motive and that their conversion was not under the guidance of the court. Hence the Torah considered it as if they were gentiles and remained forbidden. Moreover, their conduct ultimately revealed their initial intent.
Chazal explain elsewhere that Shimson was unaware of his future wives insincerity with their conversions. He was punished for his lack of effort to ascertain the sincerity of their conversation without ulterior motives.