Sources: all collections; answered in English in the Scholarly style.

The sources retrieved give only one direct statement linking Shirat HaYam to marriage; they do not explain why the connection exists.

The Segula Itself

[Sefer HaMiddot, Marriage.1]: "מִי שֶׁקָּשֶׁה לוֹ לִמְצא זִוּוּגוֹ יאמַר בְּכַוָּנָה שִׁירַת הַיָּם" — "One who has difficulty finding his match should recite Shirat HaYam with kavana." This is the sole source in the retrieved material stating the practice. No reason is given.

Kavana

[Peninei Halakhah, Women's Prayer, 1:8:8]: "It is said in the name of Arizal that kavana is like the wings upon which prayer soars heavenward and is accepted." This explains why kavana is specified: without it, the recitation lacks the quality that makes it efficacious.

Gap in the Sources

The retrieved sources do not explain the theological or kabbalistic rationale connecting Shirat HaYam specifically to finding a zivug. [Likkutei Etzot, Marriage.1] notes that intense prayer generally helps one find a partner (citing Likutey Moharan I, 9), but does not mention Shirat HaYam. [Likutei Halakhot, Orach Chaim, Laws of Passover, 8:2:1] connects Shirat HaYam to milah and the concept of Kriat Yam Suf, but not to marriage. The inner reason for the segula must be sought elsewhere.

References

1. Sefer HaMiddot, Marriage.1
2. Likutei Halakhot, Orach Chaim, Laws of Passover.8:2:1
3. The Koren Shalem Siddur; Ashkenaz, Understanding Jewish Prayer.98
4. Peninei Halakhah, Women's Prayer, .12:8:2
5. Peninei Halakhah, Women's Prayer, .1:8:8
6. Mei HaShiloach, Volume I, Exodus, Yitro.2
7. Chiddushei HaRadal on Devarim Rabbah.11:7:1
8. Malbim on Genesis.10:21:1
9. Malbim on Malachi.3:17:1
10. Ta'amei HaMinhagim, Miscellany.80:1
11. Beit Yisrael (Ruzhin), Aphorisms.7
12. Shem MiShmuel, Shavuot.11:8
13. Midrash Tanchuma Buber, Beshalach.11:4
14. Likkutei Etzot, Marriage.1
15. Machzor Rosh Hashanah Edot HaMizrach, Shacharit, Shirat Hayam.3
16. Machzor Rosh Hashanah Edot HaMizrach, Shacharit, Shirat Hayam.1
17. Siddur Ashkenaz, Shabbat, Shacharit, Pesukei Dezimra, Shirat HaYam.1
18. Machzor Rosh Hashanah Sefard, The Morning Prayers, Shirat Hayam.1
19. Machzor Yom Kippur Sefard, The Morning Prayers, Shirat Hayam.1
20. Mishneh Torah, Prayer and the Priestly Blessing.7:13
21. The Koren Shalem Siddur; Ashkenaz, Festivals, Additions to Shaharit for Yom HaAtzma'ut.1
22. Rabbi Sacks on Siddur, Festivals, Hallel.14
23. Machzor Rosh Hashanah Edot HaMizrach, Shacharit, Shirat Hayam.2