The custom is to say aloud the 13 Attributes of Mercy. The only source is written in Pirkei d’Rabbi Eliezer (end of siman 46) that Moshe cried out in a loud voice HaShem, HaShem, etc. However, no reason is advanced for this custom.
In general, our custom is not to call loudly in prayers, as evidenced from Chana’s prayers (Shmuel 1,1) that her lips moved but no sound was heard. Also, the Gemara (Berachos 24b) mentions that one who raises one’s voice during prayers is from those that lack faith (as if G-d could only hear if one is loud).
Even though we say in our prayers שמע קולנו, this is not meant to imply that our prayers are loud in order to be heard. Rather, the implication is that our prayers should be received and accepted, similar to the verse (Shmos 3, 18) ושמעו לקולך, that our words should be accepted. When the Chumash intends to emphasize an audible sound, then the word קול is expressed with an adjective, such as found in the following: קול גדול, קול אדיר, קול רם וכדומה. Therefore, שמע קולנו doesn’t (necessarily) mean loud.
The applicable source for saying aloud the 13 Attributes is from Berachos 32a. The Gemara asks why the verse (in the saying of the 13 Attributes) begins ויחל משה (and not that Moshe prayed). The word יחל teaches that Moshe stood up and prayed until the point that he reached אחילו, which means the fiery essence. And when a person feels a burning heart, one raises one’s voice and cries out. Thus, Moshe raised his voice when he said the initial 13 Attributes. Thus, when we wish to call out for mercy, we attempt to emulate Moshe and also say also the 13 Attributes.
