Gemara:


Ein Ayah: Berachot are recited by using the parts of the body that accomplish speech. However, a person needs to recognize that the physical side just enables one to express the essence, which is the internal inspiration and true recognition that come with the speech. The berachot contain in their depths lofty emotions of knowledge and fear of Hashem. These have a major impact on the individual people who make the berachot and even upon those who join together in regard to the berachot.
Answering amen demonstrates the internal goal of agreeing to and internalizing the berachot. Since the main element of amen is internal, it is important to demonstrate that the excitement that comes with external actions is not the main thing. That is why it is proper not to answer amen louder than the one making the beracha. If the answerer is louder, it shows that he feels that what he is adding is the additional excitement. It is true that this can sometimes be worthwhile, but it is not the foundation of true success but a means of reaching true realizations. Therefore, he is allowed to show his excitement physically, but he should do so in a manner that shows that it is not by answering loudly but rather by increased concentration. This concentration does not require verbalization, as it above the level of verbalization.
The point of speech is to express to others that which one is thinking. The internal success of a person is in meriting feeling the good fortune of success in one’s circumstances. The internal recognition of the beauty of truth will fill the world and be etched upon every heart.
This is the meaning behind the pasuk, "Declare Hashem’s greatness with me." One should do so in a visible manner along with others, starting with an external action and internalizing. In order for it to have impact on others it is proper that it be audible. However, "...we will extol Him together," which refers to the internal recognition of Hashem’s great honor, purity, and kindness, is best accomplished in silence. This is to be contemplated within one’s own mind after being inspired by those people of complete wisdom who informed him of the truth. This is the matter of "extolling together" in a way that the voice is secondary, and the main thing is the realization and internalization that amen demonstrates. That is the reason why amen is said as one word, when really it represents three Hebrew words (kel melech ne’eman), meaning "Hashem is the trustworthy King." The idea is that one should speak in short and think at length. For the same reason, one should not be too loud.