The Torah basically tells us the reason. Leviticus 17:10-12 says:
וְאִישׁ אִישׁ מִבֵּית יִשְׂרָאֵל, וּמִן-הַגֵּר הַגָּר בְּתוֹכָם, אֲשֶׁר יֹאכַל, כָּל-דָּם--וְנָתַתִּי פָנַי, בַּנֶּפֶשׁ הָאֹכֶלֶת אֶת-הַדָּם, וְהִכְרַתִּי אֹתָהּ, מִקֶּרֶב עַמָּהּ. כִּי נֶפֶשׁ הַבָּשָׂר, בַּדָּם הִוא, וַאֲנִי נְתַתִּיו לָכֶם עַל-הַמִּזְבֵּחַ, לְכַפֵּר עַל-נַפְשֹׁתֵיכֶם: כִּי-הַדָּם הוּא, בַּנֶּפֶשׁ יְכַפֵּר. עַל-כֵּן אָמַרְתִּי לִבְנֵי יִשְׂרָאֵל, כָּל-נֶפֶשׁ מִכֶּם לֹא-תֹאכַל דָּם; וְהַגֵּר הַגָּר בְּתוֹכְכֶם, לֹא-יֹאכַל דָּם.
And whatsoever man there be of the house of Israel, or of the strangers that sojourn among them, that eateth any manner of blood, I will set My face against that soul that eateth blood, and will cut him off from among his people. For the life of the flesh is in the blood; and I have given it to you upon the altar to make atonement for your souls; for it is the blood that maketh atonement by reason of the life. Therefore I said unto the children of Israel: No soul of you shall eat blood, neither shall any stranger that sojourneth among you eat blood.
In other words, the blood of an animal is designated for ritual purposes, so we shouldn't eat it. We see the same phenomenon by the prohibition on eating suet which is also designated for ritual use in the Temple, as earlier in Leviticus 7:23-26 it says:
דַּבֵּר אֶל-בְּנֵי יִשְׂרָאֵל, לֵאמֹר: כָּל-חֵלֶב שׁוֹר וְכֶשֶׂב, וָעֵז--לֹא תֹאכֵלוּ. וְחֵלֶב נְבֵלָה וְחֵלֶב טְרֵפָה, יֵעָשֶׂה לְכָל-מְלָאכָה; וְאָכֹל, לֹא תֹאכְלֻהוּ. כִּי, כָּל-אֹכֵל חֵלֶב, מִן-הַבְּהֵמָה, אֲשֶׁר יַקְרִיב מִמֶּנָּה אִשֶּׁה לַיהוָה--וְנִכְרְתָה הַנֶּפֶשׁ הָאֹכֶלֶת, מֵעַמֶּיהָ. וְכָל-דָּם לֹא תֹאכְלוּ, בְּכֹל מוֹשְׁבֹתֵיכֶם, לָעוֹף, וְלַבְּהֵמָה. כָּל-נֶפֶשׁ, אֲשֶׁר-תֹּאכַל כָּל-דָּם--וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא, מֵעַמֶּיהָ.
Speak unto the children of Israel, saying: Ye shall eat no fat, of ox, or sheep, or goat. And the fat of that which dieth of itself, and the fat of that which is torn of beasts, may be used for any other service; but ye shall in no wise eat of it. For whosoever eateth the fat of the beast, of which men present an offering made by fire unto the LORD, even the soul that eateth it shall be cut off from his people. And ye shall eat no manner of blood, whether it be of fowl or of beast, in any of your dwellings. Whosoever it be that eateth any blood, that soul shall be cut off from his people.
Note the prohibition on blood is repeated there in the same section.
The remaining question is why is blood prohibited in all animals and birds while suet is only prohibited in animals offered as sacrifices in the Temple? The simple answer is suet only serves a ritual role in those animals, while the blood serves a ritual role (via slaughtering) in all animals and birds. (One could even argue that what we think of as ordinary slaughtering outside the Temple is really a type of quasi-Korban, cf. Deuteronomy 14.)