But a non-inflexed language seems less complex than a very strongly inflexed one.

This view is not itself true, and so we need to deal with it before moving on to the rest of your question.

Language complexity remains a debated topic in Linguistics. For a long time there was a traditional view that certain languages, e.g. Latin and Greek, were more 'complex', which reflected the sophistication of their cultures. This kind of view was common in the 19th century.

From about the 1950s onwards, linguists started to reject this view, and embrace 'equal complexity' - the idea that all languages are 'equally' complex, just in different ways.

This idea itself, at least in its cruder forms, has been attacked in recent ideas. Although not reviving the older view, this newer view holds that languages can and do differ in complexity, but this doesn't say much about how useful, versatile, flexible they are for expressing ideas. (And says very little, likewise, about the 'development' of the culture associated with it).

So, let's turn to the Indo-European group. It's a very common view that fusion inflectional languages, such as Latin and Greek (mainly because of their historic role and socio-cultural perception in today's west), are 'more complex'. This is not really true, I would argue. That is, inflection only seems more complex to someone used to speaking a fairly uninflected language.

In terms of morphological typing, that is, classifying languages based on how they divide up words, there are two main categories, with one of those divided again:

Analytic - Synthetic (Agglutinative) - Synthetic (Fusional)

(though, this strict division is itself subject to critique in modern linguistics. No language is 'purely' one or the other)

Analytic languages don't inflect much. A word is a morpheme. Things like syntax (word order) and auxiliary words do the work that other languages do with endings. English is a great example of an analytic language. It has pretty low 'complexity' in terms of endings, but it has pretty 'high' complexity in terms of syntax. (Ask someone from a highly synthetic language how they felt learning English word order rules).

Agglutinative languages use more add-ons: suffixes, prefixes, and infixes. Mongolian comes to my mind. It has 8 (or so) cases for nouns, they are all formed by adding suffixes to words. Word order is more free, because those endings carry quite a bit of information.

Fusional languages are those like Latin and Greek, where there are endings (or beginnings, or middles) that change, but they change in such a way that there is not a strict root/ending division - the endings (or other elements) alter the stem itself. Hence, 'fusional'.

Languages tend to evolve over the long term from one of these categories to another. In IE languages, as in your question, you have generally seen a movement over time from Fusional towrards Analytic. But languages also move from Agglutinative to Fusional, or Analytic to Fusional. RMW Dixon has suggested that Egyptian has gone through all three over its extensive lifespan.

Back to your initial question then.

1. Yes, it's kind of true. That is, fusional languages appear to generally evolve towards analytic languages. But this isn't about complexity. Analytic languages exhibit complexity in other ways.

2. The 'why' question is much more difficult. Language change is a complex phenomenon without simple answers, so I won't venture one here.