Rashi: It is when Nebuchadnetzar authorized Tzidkiyah over the five kings (refer to 27:3:1:1). Yirmeyah said his mission, and sent the straps and sticks [of the yoke] to the kings via their messengers.

Rashi: It was the beginning of his rule over the five kings. That year he went to greet Nebuchadnetzar in Bavel, and Serayah Sar ha'Menuchah went with him (51:59).

Radak citing Seder Olam 25: That year he and Serayah Sar ha'Menuchah went to Bavel to greet Nebuchadnetzar in Bavel, and he returned to Yerushalayim to his kingship. Therefore it is considered the first year of his kingship, and it says "b'Reishis Mamleches Tzidkiyah."

Radak: We can say that it was truly Tzidkiyah's first year. It was the fourth year of the Shemitah cycle. His fourth year was the year after Shemitah, and also his last (11th) year, in which the Churban came. This is like Chazal said (Ta'anis 29a), that the Churban was in Motza'ei Shemitah. It counts from the Shemitah cycle to hint that Yisrael Yishmetu (will cease from) their inheritance because they did not observe Shemitah. It mentions the fifth month, to hint that the Churban will be in the fifth month.

Malbim: It was the fourth year of Yehoyakim. In that year, Nebuchadnetzar became king, and appointed Tzidkiyah (a) to be second to Yehoyakim, and (b) to reign after Yehoyakim dies. The kings' messengers came to Tzidkiyah, for also he dealt with royal affairs. Nebuchadnetzar appointed the five kings to be under Tzidkiyah's rule. Even though the [11] years of Tzidkiyah's reign are counted only from when Yehoyakim died, and Tzidkiyah ruled alone, he was already called king from when Nebuchadnetzar appointed him. This is why Nebuchadnetzar exiled Yechanyah - he was angry at him for reigning after Yehoyakim died. This deviation from Nebuchadnetzar's orders was a rebellion! This is why it says "va'Yimloch Melech Tzidkiyah" (37:1) - he was already king before he ruled by himself.