Cycling is not an inherently political thing. However, like many other things it is often framed in terms of the left-right political spectrum. In those who are more right-leaning, there is a tendency to frame things in terms of individualism and to look inwards to find the source of social problems, often suggesting they are a result of a character flaw or behaviour. In those who are more left-leaning, there is a tendency to frame things in terms of collectivity and centralisation and to look outwards to find the source of social problems, often seeing behaviours as a result of environmental factors or a shortcoming of a society itself. As with most things, reality probably lies somewhere between the two extremes.
But what does all this mean for cycling? From a right-leaning perspective the bicycle fits nicely with individualism, an individual mode of transport which has no negative impact on the lives of others and offers the user complete freedom of movement. In a small way it allows its users to directly act against concerns such as climate change without the need to restrict the freedom of others. The economic benefits of cycling are also particularly desirable from a right-leaning perspective. From a left-leaning perspective the bicycle fits well with collectivism by producing wide-ranging societal and environmental benefits, especially in a country like the UK where the cost of healthcare is paid for through the government. Cycling (and affordable public transport) in place of catering primarily for more expensive environmentally and socially destructive forms of transport make mobility accessible to all members of society whilst increasing the health and safety of citizens, which is extremely desirable from a left-leaning perspective.
There are downsides too. There is a tendency towards xenophobia towards the right-end of the political spectrum; where cyclists are seen as a minority group they may be assaulted, threatened & intimidated by those who see cyclists as a group being 'different' somehow from themselves. The focus on social problems being a result of character flaws means that measures to promote cycling may end up being restricted to 'awareness' and 'training,' with infrastructural changes to the built environment to benefit cyclists (and other vulnerable road users) being shunned in favour of encouraging behavioural change indirectly, which is less successful. At the left-end of the spectrum there is a tendency towards over-regulation of the behaviour of the population, with measures such as compulsory cycle helmet legislation considered despite the damage these laws are already known to do to cycling rates.
As stated at the start, cycling is not inherently a political thing. The perception of the activity and those participating is largely due to existing personal experiences or prejudices, although how these experiences or prejudices manifest themselves may be framed depending on the political persuasion of the individual. However, when talking cycling with those in power, it is worth considering how to frame cycling as a 'good thing' in terms of the political mindset of the person whom you are addressing. For example, some might consider cycle infrastructure to be an inherently left-wing means of encouraging cycling, but it could just as easily be considered in more individualistic terms as a means of extending the freedom of movement for the individual, or as a means to produce an economic benefit. It's all a matter of spin.