This is a quite complex history, because of Indonesia is located in the middle of trade routes between India/Middle East (mostly Arabs) and China or East Asia, and therefore the language has evolved and influenced by many other languages.

But as far as I know, the language of Malay is in the same root as those of Polynesian family, you know those island along the Pacific ocean. After the break up of these island thousands of years ago, they developed into its own distinctive language but since the roots are the same, you can almost see that there are similarities in Bahasa Indonesia, Malay (spoken in Malaysia), Philipino Tagalog, and several other Polynesian islands.

Indonesia itself, as you know consists of approximately 13,677 (or is it 16,377 ?) islands, it's largest island nation in the world, and I believe there are more smaller, insignificant islands. It is known that there are 300 different dialects spoken by the different ethnic groups on the major islands. These people have totally different culture, ethnicity, cuisine, and believes. Within the dialect itself, there are many variations in different areas.

However, it was the Malay language, which is a language based in West Sumatra, Riau, Palembang area on the Sumatra island, also spoken in Malaysia but slightly different (and the entire Mallaca Straits, before Singapore becomes a nation) that was evolved to be the National Language of Indonesia. Kind of like Mandarin becomes the National Language of China. Bahasa Indonesia (Indonesian language) is the modern version of Malay. When you compare Malay and Indonesian, it's like comparing the modern American English and Brittish English.

I don't quite remember how the Indonesian chooses Malay to be its national language, I assume Jojon is somewhat correct, there used to be a powerful Kingdom called Sriwidjaya, one of the richest, most influentual kingdom in the Asia in the Palembang region and they adopted Malay (obviously) as their language, and as they expanded and become more prosperious, Malay becomes the common language, especially if you want to trade along the coast.

Most of Classic Indonesian literature are also in Malay, and most of their classic authors are those who were born and raised in Padang/Palembang/Riau area. These people are famous throughout the land for having the natural talent with poetry and literature. Many of their classical novels and poetries have become Indonesian Classics, required for students to learn in elementary/middle schools.

However, as I said earlier, Indonesia prospered with trades with the Indians, Arabs, and the Chinese. Over the thousand of years, there are many words from India and China assimilated into Indonesian language.

When Islam prospered in the Arab land, their traders come to Indonesia to rest on their way to China to trade porcelin, silk, paper, and introduced their religion to Indonesia. Thus, the highly restricted Hinduism and Buddishm influence become weak.

Why ? Because the Moslems encourage equality, which is a big thing. Indonesia was a highly Hinduism/Buddhism influenced country, but its powerful Kingdoms including Sriwidjaya and Majapahit implemented the idea of the caste systems. But the Moslems provide more prosperity and introduce the idea of equality.
With this new waves of Moslem traders, Islam spread inland from around the Malacca straits, as a result, the Buddhist and Hindus got cornered in Java and the majority of Hindus finally moved into an island called BALI.

The Language was heavily influenced with Arabs after this, and the waves of European Colonialization followed.

First it was the Portugese, then the Spanish, and then the Dutch. The Dutch occupied Indonesia for approximately 350 years and the language again added more vocabulary from the Dutch. Indonesia, however, for a brief period in between was also occupied by the British, and some English words played along in Bahasa Indonesia. Then finally, comes the Japanese, the Japanese too added some words into Inonesian language.

For example,

The word Sepatu (means shoes in English) is from a Spanish word, Sepatos. The word bola (means ball in English) is from an English word, ball. The word lihai is from a Chinese word means just exactly the same lihai - witty, smart, talented, able. The word teh (means tea) is from an English word Tea.

I'm not sure when was the official date, Indonesian (Modern Malay) was enacted as the official language ? Anyone want to fill in ?

After the Independence on August 17, 1945, Indonesian have been widely spoken just about everywhere in the archipelago, and many newspapers were published using Bahasa Indonesia. However, the language have been modernized and simplified on many occasions ever since.

For example Jakarta used to be spelled Djakarta