How are borrowings in local currency used to hedge foreign exchange exposure on investments?

The phrase 'local currency' in the quoted section of your question refers to currency local to the same jurisdiction as the company's individual foreign investments. They are referring to the idea of borrowing funds in the same currency as where the revenue will come from.

Consider a US mega-corp with holdings around the world. Something that requires heavy investment wherever it expands operations, where it sells into the same local markets, meaning revenue for different operations is in these foreign 'local' currencies*. A good example would be manufacturing, in a case where the manufactured goods are expected to be sold locally.

You have two options to pay for this expansion, assuming you need to take on debt to do so: you could borrow your 'home' USD currency, and then sell it for CAD or whatever to pay local employees, or you could directly borrow CAD. Assume that for the first 5 years, total debt repayment [principal repayments + interest costs] on debt might eliminate all or nearly all of the anticipated revenue. This means that failing to put the debt repayment costs in the same currency as the anticipated revenue, can create a large fx risk.

If you have $100M in CAD revenue and $100M CAD in debt repayment, then you have 0 net USD cashflow, regardless of the fx rate. In 5 years, assume you pay off the debt, and you have $100M in CAD revenue. It is true that the USD-equivalent value of that cashflow will be volatile according to the exchange rate, but less so than a case where inflows and outflows are in different currency. ie: volatility in value of CAD, between 0.75 USD and 0.85 USD means that your net income in year 5 would be between 75M USD and 85M USD [ie: your net income has volatility of like 10-15%].

If you have $100M in CAD revenue but $80M in USD debt repayment, then a strengthening CAD gives you higher net cashflow, and a weakening CAD gives you lower net cashflow. If the value of CAD moves between .75 USD and .85 USD, then you could have anything from a loss of ~$5M USD to a gain of $5M USD [ie: your net income has volatility of like -100% to +100% This is proportionately a much larger degree of volatility than a shift of 10-15% as in the above 'hedged' example]. Once you pay down the debt, the risk is the same as above, but in the interim period, the risk is greater where the cost of debt is not offset naturally by the incoming revenue.

Matching your cash inflows and outflows in the same currency naturally offsets some of the fx risk associated with foreign expansion, without needing to undertake potentially more expensive hedging processes.

*Note that in some cases, location of the foreign investment won't impact revenues - for example in Oil & Gas, revenues would almost always be in USD by industry convention, so in a case like that, the relevant revenue currency to hedge with the correct debt currency would be USD [which for accounting purposes would likely be considered the 'local currency' anyway, and would be grouped as such in the broad statement in your question].