Hopefully someone will provide a better answer than this, but this should temporarily suffice. As far as I am aware, we have very little extant sources from the Zheng He period aside from the travelogues of Ma Huan. I can offer you a guess based on inductive reasoning, however.

According to Sanbao Taijian Xia Xiyang Ji Tongsu Yanyi (a historical novel by Luo Maodeng in 1597), there was a subclass of ship called horse ships (“ma chuan”). These ships, according the Ming Huidian (Ming Statutes), these ships were used for transport of cavalry and the supporting people and supplies. Because of the nature of these ships, with facilities available for horses, my guess would be that this ship could be modified to suit the needs of a giraffe.

However, the giraffe in question (seen in now-famous painting of its presentation with a handler), seems to actually have been brought independent of Zheng’s trade mission as its arrival at the imperial capital was recorded in the Ming Shilu (Veritable Records of the Ming), in 1414 while Zheng’s fourth voyage was still underway (it would return in 1415). Sally K. Church believes that this means the Bengalis who presented the giraffe made their own expedition to the Ming to present the animal as tribute. (Sally K. Church, “The Giraffe of Bengal: a Medieval Encounter in Ming China” The Medieval History Journal 7, 1 (2004))

So while my answer has only really guessed at it from knowledge of the logistics of the time, it seems more likely that someone other than Zheng may have transported the initial giraffe. I do not know how the subsequent animals arrived as well. While the Bengali envoys that brought it likely had their own methods, the Ming were certainly capable of carrying large quadrupeds inside their large (though still smaller than the treasure ships) Horse Ships. If someone knows further about the historical weather and sea patterns of the routes they took (my understanding is that they were quite calm), this could further help reason out the potential answer, but I am afraid it is pretty unlikely that there is a cut and dry answer that says “here is the method.” Now I am curious about it myself!