Great question! This is certainly something a lot of candidates experience, and learning to feel comfortable with networking is something you will have to navigate as a consultant. My advice would be to approach the interaction with curiosity and openness, and ‘fake it till you make it’ if you don't feel confident yet.
People get these messages asking to network all the time - it is perfectly normal, and most people enjoy having those kinds of conversations and giving others advice! So don't feel awkward about doing it, no one is going to find it strange or transactional, and the worst thing that can happen is you get no response or someone tells you they are a little too busy to talk right now. Send many of these messages, and ask to talk to a wide range of people.
In your message, you can start by mentioning any connections you have with the person (e.g., if you are interested in the industry they work in, or went to the same school) to establish common ground, and then just ask them for some time to talk. Approach the conversations with an outlook to getting to know them, learning more about their work and day-to-day life, rather than a transaction you have to engage in to get a referral. I think asking for suggestions of other people who you could reach out to (e.g., who work in the industry) can be a nice way of starting to develop a network in a more organic way.
At the end of the day, networking feels awkward for most people to begin with. However, you get better and more natural at it with practice, so the best advice on this is just to go for it! You will get more comfortable with it over time, but as long as you are polite and professional, and curious and open in your interactions, you can't really go wrong.