Rav Shimshon Raphael Hirsch, in his Torah commentary, explains the symbolic significance of this requirement of tamim, which demands both external and internal perfection:

"So that for each and every aspect of our relation to G-d which is to be expressed by an offering, "completion," i.e. the demand for the application of the whole of oneself to that aspect, is the first and most indispensable condition... Anything missing of the living creature which is to express our relation to G-d would give the idea of excluding this missing organ, and thereby the sphere of our life which this organ serves to represent, from our relation to G-d, and from our giving ourselves up to Him in that particular sphere...so certain is it that the very smallest, the slightest keeping back of any part of ourselves and our lives as being excluded from our relation to G-d is a denial of His Oneness."

The sacrificial offering represents the concept of our devotion to Hashem.  The absence of any organ, whether external or internal, would imply that the aspect of our being which is signified by that organ is excluded from this devotion.  The requirement of tamim thus relates to the all-encompassing nature of avodat Hashem, the obligation we bear to devote ourselves entirely to the service of G-d.  Just as every part of the animal must be present and intact for it to qualify as a sacrifice, so must each and every part of our body and soul be included in our religious devotion.  

We all have unique talents and traits that help us relate to our service of Hashem. For some of us, religion is intellectual, as we explore the depths of the Torah and nuances of Halacha with our brains. For others it is emotional, as we search for spiritual heights within the depths of our "hearts." And for others it is through our arms as they shake a lulav, our hands when they are used to save lives, or our eyes when they are used to seek out the the less fortunate among us. While all these strengths are necessary,  it is important that we focus on being well-rounded as well, to ensure that every part of our body and soul is included in our devotion to Hashem.