Primarily, this is because in the modern secular worldview, religion itself is seen as outdated or, at most, something that should be kept private. In contrast, Islam is a heavily communal and public religion that involves both public and private life.

Also, there are some aspects of the classical Islamic worldview that do not match the current world. For instance, in classical Islamic texts, identity is primarily treated via religious identity, whereas today, one's nationality is considered the primary aspect of one's identity. So there have been some shifts in the world.

Furthermore, one aspect of secular modernity is the belief that human society is always improving, and that modern times are better because we have more scientific discoveries and technology. That is to say, the new is always better than the old, and so anything pre-modern must necessarily be inferior. This belief is perhaps waning, in light of increasing awareness of global inequity surrounding the production of technology, and also environmental concerns, but is not uncommon. 

Perhaps one can also add that after Europe and the West came to power (culturally, politically, economically, and scientifically), a tendency to look down on Islam as backwards developed. This was not found in previous centuries, when Muslim civilization was culturally and scientifically ascendant. 

So these are some reasons why some people see Islam as backwards, although individual people would have to say their own reasons.

It is worth noting that all of these views relate to trends of thought among people today and are not absolutes; for instance, in some eras, classical literature and ideas have been valued, and there wasn't the idea that new things are necessarily better than old things.

As for what to do to counter it, first, one can only control other people's views so much. Although it is good to stand up for one's religion, in the end sometimes the most powerful statement is simply having confidence in your faith and living it. Islam has been a counter-example to the idea that religions will become irrelevant in modernity, because many Muslims hold quite strongly to Islamic teachings in the modern era or in Muslim-minority societies, especially Western countries. When this is seen on a mass scale, it proves that Islam has the ability to flourish in modernity and to provide an alternate path to modernity than secularism.

Also, it is good to keep in mind that every generation has its own ideas of what is new or outdated, but those ideas change quickly. 40 or 100 years ago, some people thought they knew everything, and then came to realize that there were things they were mistaken about, such as quantum physics, the effects of some chemicals, and so on. 

An underlying idea in Islam is that while societies come and go, the underlying human nature does not change, nor does the process of spiritual growth of the human being, the spiritual trajectory of the human being, or the things which nurture the human being versus the things that harm it (that is to say, the halal and haram). It is narrated that human beings from the time of Adam (A) to the end are like teeth on a comb, that is, we are all essentially the same. So, from that angle, the teachings of Islam are relevant in any era. 

Of course there are some matters which are new in Islamic thought, such as how to pray in space, or questions about democracy and political matters, which do relate to new issues and not past issues, and which are discussed via the Islamic sources as well. So from that angle there is room for growth in Islamic thought without deviating from the ideals of the teachings of the Prophet (S).

Lastly, one has to admit that some Muslims hold ideas which could be called "backwards" and ascribe them to Islam. They usually do not have anything to do with the teachings of the Qur'an or Prophet (S) but rather are their own ideas, or else are some very limited interpretations of the Qur'an and hadith. Given that there are over a billion Muslims, it is natural that there is going to be some variation of ideas and some ideas might seem "backwards" (i.e. wrong or unenlightened). Perhaps the biggest form of backwardsness is when some Muslims, perhaps due to feeling defensive in the face of opposition to Islam, do not wish to engage with the intellect in open discussion about Islam, as Allah frequently tells us in the Qur'an to think, reflect, and so on. However, this sometimes results in narrow thinking.

It is a big question - hope that helps! Some others may have some insights as well.