One of the most powerful relationship-building tools I have found in my career is the simple act of asking for advice. It's also something that I don't see a lot of people take advantage of, especially in tech. In some cases, it's an ego thing. What if they think I'm weak for not knowing, for not figuring this out on my own? Sometimes, it's a desire not to be a bother to the other person. Sometimes it just doesn't occur to you to seek outside help. But when you don't ever ask others for real advice (more than simply, "how does this code work anyway?", or, "what was that cool hack you cooked up the other day?") you leave a lot of potential on the table.
It shows you respect the other person's opinion (and thus, the other person)
We all want to believe that our opinion is important, and when you are actually asked for your opinion it is a token of respect for you as a person. You've heard that people like those that like them back? People tend to respect those that respect them back. After all, you have such good judgement in respecting them!
This really comes to play when you ask people that work for you or are junior to you for their advice, especially when you follow it or use it to open a dialogue on a bigger topic. As a technical leader this will usually come in the form of advice about technology, and let's be honest: none of us has the time to be a true expert in all of the technologies we use in our job. Why would you hoard the decision making on all technical matters? Your employees probably come to you with suggestions and opinions already, but taking care to specifically ask them for their advice is an important way to ensure they know that you respect them and the knowledge they posses.
It makes them a partner in your success
Think about a time when you've helped someone solve a particularly hard bug they were facing. A bug that was meaty and fun and made you think and also made you feel awesome and smart for being involved in solving it. Afterwards, did you feel some ownership stake in the success of that project? This is the feeling you give people when you ask them for their advice, especially when you take it and turn it into success.
Now think about a time when you've hit a bug in someone else's code and they have ignored your attempts to help them debug it, or worse, totally ignored your patches and pull requests. You probably feel a bit more animosity towards the project, and if it fails, perhaps it deserves to fail.
This set of feelings is analogous to proactively going to your manager and asking for advice on how to grown in an area, versus having to be told by your manager that you are failing. In the first scenario, you make your manager a partner in your success, and they feel pride and sense of ownership when you succeed. In the second case, you are at best an area for improvement and a worst a burden or even a write-off.
It forces you to think about the problem you really have
You shouldn't ask someone for advice without knowing what you're looking for. Just as you're not likely to get much help debugging if all you know is that the program crashes, simply asking someone "Hey, any advice?" will probably get you little better than a cliche like... "ask for advice". Really valuable advice doesn't come from emailing Kevin Systrom and asking how you can get a billion dollars by putting your Hadoop cluster in a Cray-1. It's delivered when you tell your boss you're struggling with the challenge of managing a team and writing code at the same time, and how does he manage to keep an eye on everyone without micromanaging or spending every waking hour watching GitHub and Jira?
Of course, this comes with some stipulations. The act loses its power if you're just doing it for the sake of doing it. You have to be willing to at least consider taking the advice you are given, and you don't want to waste people's time. But if you take the time to truly think about areas where you'd like to improve, and go to people honestly asking for advice on how to do so, you may find that an extra set of eyeballs makes the bugs in your career shallow.